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Estimation

linear-demand linear-cost, \(\sigma = 1\)

\(\theta = 0.05\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.817 1.000 7.959 9.580 15.288
alpha1_hat 100 0 0.825 0.857 −1.470 0.766 3.413
alpha2_hat 100 0 0.110 0.081 −0.105 0.109 0.348
alpha3_hat 100 0 0.979 0.217 0.494 0.962 1.691
gamma0_hat 100 0 2.114 0.966 −0.165 2.044 4.304
gamma1_hat 100 0 1.873 8.799 −14.800 0.921 80.625
gamma2_hat 100 0 0.881 0.150 0.497 0.890 1.265
gamma3_hat 100 0 0.894 0.174 0.512 0.885 1.249
theta_hat 100 0 −0.693 5.200 −47.239 −0.098 10.254
theta_hat_t_stats 100 0 −0.292 1.122 −3.414 −0.214 2.518
R2_demand 100 0 0.370 0.292 −1.623 0.433 0.715
R2_supply 100 0 0.981 0.006 0.963 0.981 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.978 0.748 8.375 9.847 12.592
alpha1_hat 100 0 0.829 0.670 −0.830 0.745 2.707
alpha2_hat 100 0 0.116 0.061 −0.056 0.113 0.304
alpha3_hat 100 0 0.999 0.167 0.570 0.990 1.613
gamma0_hat 100 0 1.422 1.007 −1.241 1.554 3.329
gamma1_hat 100 0 0.772 1.900 −10.596 1.012 4.237
gamma2_hat 100 0 0.952 0.158 0.634 0.941 1.325
gamma3_hat 100 0 0.953 0.144 0.638 0.933 1.270
theta_hat 100 0 0.094 0.943 −1.810 0.018 5.169
theta_hat_t_stats 100 0 −0.020 0.937 −1.932 0.037 2.466
R2_demand 100 0 0.325 0.241 −0.787 0.387 0.644
R2_supply 100 0 0.977 0.005 0.966 0.977 0.986
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.973 0.410 9.143 9.948 10.873
alpha1_hat 100 0 0.947 0.358 0.030 0.932 1.842
alpha2_hat 100 0 0.104 0.034 0.028 0.103 0.204
alpha3_hat 100 0 0.998 0.103 0.757 0.985 1.293
gamma0_hat 100 0 1.116 0.720 −0.621 1.194 2.685
gamma1_hat 100 0 1.019 0.896 −2.048 0.918 3.368
gamma2_hat 100 0 0.982 0.104 0.767 0.981 1.220
gamma3_hat 100 0 0.991 0.097 0.823 0.980 1.246
theta_hat 100 0 0.022 0.462 −1.450 0.084 1.473
theta_hat_t_stats 100 0 0.083 1.033 −2.923 0.275 2.766
R2_demand 100 0 0.332 0.127 −0.047 0.341 0.540
R2_supply 100 0 0.975 0.005 0.960 0.976 0.986
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.997 0.199 9.646 9.993 10.439
alpha1_hat 100 0 0.995 0.169 0.621 0.983 1.359
alpha2_hat 100 0 0.101 0.018 0.067 0.101 0.140
alpha3_hat 100 0 1.001 0.046 0.901 0.994 1.106
gamma0_hat 100 0 1.037 0.314 0.129 1.049 1.743
gamma1_hat 100 0 1.013 0.298 0.231 1.025 2.007
gamma2_hat 100 0 0.996 0.050 0.887 0.991 1.143
gamma3_hat 100 0 0.996 0.042 0.873 0.997 1.101
theta_hat 100 0 0.037 0.159 −0.497 0.028 0.504
theta_hat_t_stats 100 0 0.213 0.955 −2.446 0.178 2.812
R2_demand 100 0 0.342 0.050 0.237 0.340 0.458
R2_supply 100 0 0.975 0.002 0.969 0.975 0.980
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.996 0.146 9.640 9.993 10.428
alpha1_hat 100 0 1.012 0.118 0.683 1.016 1.274
alpha2_hat 100 0 0.099 0.011 0.073 0.099 0.124
alpha3_hat 100 0 1.000 0.037 0.909 1.002 1.081
gamma0_hat 100 0 1.063 0.204 0.574 1.071 1.535
gamma1_hat 100 0 0.995 0.231 0.485 0.987 1.488
gamma2_hat 100 0 0.993 0.031 0.920 0.991 1.078
gamma3_hat 100 0 0.997 0.026 0.928 0.997 1.056
theta_hat 100 0 0.043 0.123 −0.246 0.050 0.289
theta_hat_t_stats 100 0 0.371 1.060 −1.886 0.431 2.479
R2_demand 100 0 0.332 0.041 0.221 0.332 0.431
R2_supply 100 0 0.975 0.001 0.972 0.975 0.977
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.080 9.808 10.004 10.179
alpha1_hat 100 0 1.000 0.075 0.819 0.995 1.214
alpha2_hat 100 0 0.100 0.007 0.080 0.101 0.117
alpha3_hat 100 0 1.000 0.021 0.950 0.999 1.063
gamma0_hat 100 0 1.013 0.153 0.632 1.023 1.385
gamma1_hat 100 0 0.971 0.126 0.583 0.976 1.254
gamma2_hat 100 0 0.998 0.021 0.937 0.998 1.044
gamma3_hat 100 0 1.000 0.020 0.964 0.999 1.061
theta_hat 100 0 0.063 0.070 −0.096 0.064 0.301
theta_hat_t_stats 100 0 0.858 0.948 −1.404 0.910 3.819
R2_demand 100 0 0.336 0.022 0.287 0.336 0.389
R2_supply 100 0 0.974 0.001 0.972 0.975 0.977

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.059 1.165 7.650 9.738 14.910
alpha1_hat 100 0 1.176 1.935 −3.982 1.278 7.462
alpha2_hat 100 0 0.490 0.207 0.124 0.465 1.534
alpha3_hat 100 0 0.985 0.246 0.477 0.942 1.737
gamma0_hat 100 0 1.947 1.116 −1.608 2.148 4.420
gamma1_hat 100 0 0.690 2.605 −6.217 0.917 10.529
gamma2_hat 100 0 0.896 0.190 0.456 0.889 1.588
gamma3_hat 100 0 0.902 0.187 0.388 0.889 1.384
theta_hat 100 0 −0.026 0.457 −1.232 −0.039 0.868
theta_hat_t_stats 100 0 −0.171 1.102 −3.503 −0.090 1.785
R2_demand 100 0 0.458 0.301 −1.023 0.537 0.783
R2_supply 100 0 0.973 0.007 0.946 0.973 0.989
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.954 0.651 8.636 9.896 11.753
alpha1_hat 100 0 1.072 1.230 −1.860 1.047 3.562
alpha2_hat 100 0 0.486 0.131 0.170 0.479 0.855
alpha3_hat 100 0 1.000 0.157 0.639 0.996 1.408
gamma0_hat 100 0 1.401 0.805 −0.700 1.387 2.797
gamma1_hat 100 0 0.892 1.392 −2.129 0.843 5.739
gamma2_hat 100 0 0.960 0.120 0.704 0.965 1.335
gamma3_hat 100 0 0.955 0.130 0.636 0.942 1.339
theta_hat 100 0 0.012 0.251 −0.884 0.016 0.472
theta_hat_t_stats 100 0 −0.017 0.936 −2.551 0.071 1.758
R2_demand 100 0 0.513 0.142 0.085 0.535 0.744
R2_supply 100 0 0.968 0.007 0.946 0.969 0.981
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.044 0.566 8.881 10.032 11.537
alpha1_hat 100 0 1.174 1.117 −1.140 0.994 4.438
alpha2_hat 100 0 0.489 0.121 0.227 0.496 0.754
alpha3_hat 100 0 1.002 0.118 0.800 0.995 1.399
gamma0_hat 100 0 1.255 0.681 −0.633 1.279 2.945
gamma1_hat 100 0 1.010 1.065 −1.901 1.190 3.266
gamma2_hat 100 0 0.973 0.105 0.714 0.968 1.268
gamma3_hat 100 0 0.976 0.106 0.729 0.970 1.279
theta_hat 100 0 0.024 0.186 −0.336 −0.001 0.525
theta_hat_t_stats 100 0 0.062 0.907 −2.018 0.001 1.902
R2_demand 100 0 0.510 0.116 0.133 0.530 0.706
R2_supply 100 0 0.967 0.005 0.953 0.967 0.979
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.016 0.175 9.584 10.020 10.482
alpha1_hat 100 0 1.003 0.360 0.058 0.995 1.761
alpha2_hat 100 0 0.501 0.041 0.418 0.497 0.596
alpha3_hat 100 0 1.004 0.044 0.928 1.001 1.117
gamma0_hat 100 0 1.049 0.261 0.426 1.039 1.809
gamma1_hat 100 0 1.019 0.435 0.019 0.997 2.187
gamma2_hat 100 0 0.993 0.041 0.876 0.993 1.110
gamma3_hat 100 0 0.990 0.043 0.873 0.990 1.109
theta_hat 100 0 0.041 0.080 −0.201 0.042 0.212
theta_hat_t_stats 100 0 0.473 0.976 −2.353 0.525 2.507
R2_demand 100 0 0.519 0.039 0.418 0.519 0.600
R2_supply 100 0 0.965 0.002 0.958 0.964 0.969
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.012 0.137 9.681 9.998 10.394
alpha1_hat 100 0 0.991 0.228 0.396 1.004 1.556
alpha2_hat 100 0 0.503 0.027 0.432 0.503 0.568
alpha3_hat 100 0 1.005 0.031 0.929 1.009 1.097
gamma0_hat 100 0 1.031 0.223 0.388 1.008 1.633
gamma1_hat 100 0 0.972 0.343 0.102 0.972 1.821
gamma2_hat 100 0 0.997 0.035 0.921 0.999 1.117
gamma3_hat 100 0 0.996 0.034 0.926 0.991 1.087
theta_hat 100 0 0.049 0.061 −0.082 0.050 0.194
theta_hat_t_stats 100 0 0.852 1.050 −1.473 0.869 3.347
R2_demand 100 0 0.520 0.031 0.432 0.520 0.580
R2_supply 100 0 0.964 0.002 0.960 0.964 0.969
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.089 9.823 10.002 10.274
alpha1_hat 100 0 1.009 0.192 0.503 1.023 1.425
alpha2_hat 100 0 0.500 0.019 0.454 0.501 0.557
alpha3_hat 100 0 0.999 0.019 0.947 1.001 1.035
gamma0_hat 100 0 0.994 0.119 0.748 0.991 1.304
gamma1_hat 100 0 1.002 0.184 0.340 0.994 1.374
gamma2_hat 100 0 1.000 0.018 0.956 1.001 1.041
gamma3_hat 100 0 1.002 0.020 0.956 1.001 1.046
theta_hat 100 0 0.051 0.035 −0.036 0.055 0.152
theta_hat_t_stats 100 0 1.406 0.964 −1.038 1.481 4.087
R2_demand 100 0 0.519 0.021 0.463 0.522 0.557
R2_supply 100 0 0.964 0.001 0.962 0.964 0.967

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.933 0.897 7.453 9.922 12.191
alpha1_hat 100 0 0.943 3.335 −7.741 1.125 8.999
alpha2_hat 100 0 0.990 0.372 −0.355 0.966 1.926
alpha3_hat 100 0 0.992 0.218 0.563 0.996 1.477
gamma0_hat 100 0 1.772 1.204 −1.645 1.809 4.619
gamma1_hat 100 0 1.545 3.887 −13.337 1.151 14.505
gamma2_hat 100 0 0.929 0.181 0.523 0.917 1.512
gamma3_hat 100 0 0.938 0.204 0.610 0.904 1.641
theta_hat 100 0 −0.098 0.488 −2.143 −0.086 1.998
theta_hat_t_stats 100 0 −0.428 1.101 −3.157 −0.282 1.937
R2_demand 100 0 0.599 0.163 −0.154 0.641 0.834
R2_supply 100 0 0.967 0.010 0.936 0.969 0.984
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.938 0.640 8.767 9.912 11.649
alpha1_hat 100 0 1.277 2.296 −3.572 1.307 9.090
alpha2_hat 100 0 0.957 0.243 0.171 0.962 1.500
alpha3_hat 100 0 0.996 0.141 0.591 0.976 1.348
gamma0_hat 100 0 1.311 0.979 −1.810 1.326 3.445
gamma1_hat 100 0 0.622 2.459 −6.822 0.617 6.877
gamma2_hat 100 0 0.967 0.143 0.607 0.960 1.338
gamma3_hat 100 0 0.948 0.144 0.610 0.957 1.305
theta_hat 100 0 0.047 0.252 −0.462 0.029 0.706
theta_hat_t_stats 100 0 0.072 1.000 −2.681 0.095 2.783
R2_demand 100 0 0.562 0.128 0.062 0.583 0.757
R2_supply 100 0 0.964 0.010 0.919 0.965 0.983
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.499 8.849 9.898 12.162
alpha1_hat 100 0 1.156 1.481 −2.597 1.140 5.041
alpha2_hat 100 0 0.986 0.184 0.567 0.987 1.489
alpha3_hat 100 0 0.997 0.102 0.805 0.980 1.371
gamma0_hat 100 0 1.219 0.681 −0.445 1.254 2.900
gamma1_hat 100 0 1.046 1.521 −3.369 0.897 4.674
gamma2_hat 100 0 0.982 0.099 0.795 0.979 1.237
gamma3_hat 100 0 0.979 0.100 0.781 0.973 1.318
theta_hat 100 0 0.014 0.148 −0.382 0.008 0.416
theta_hat_t_stats 100 0 0.069 0.883 −1.915 0.049 2.307
R2_demand 100 0 0.570 0.128 −0.289 0.581 0.734
R2_supply 100 0 0.962 0.007 0.941 0.963 0.978
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.006 0.190 9.664 10.007 10.476
alpha1_hat 100 0 1.074 0.713 −0.916 1.075 3.064
alpha2_hat 100 0 0.994 0.084 0.760 0.989 1.234
alpha3_hat 100 0 0.998 0.048 0.896 0.995 1.125
gamma0_hat 100 0 1.078 0.278 0.252 1.073 1.662
gamma1_hat 100 0 0.924 0.642 −0.820 0.943 2.602
gamma2_hat 100 0 0.990 0.042 0.873 0.988 1.109
gamma3_hat 100 0 1.003 0.040 0.912 0.998 1.106
theta_hat 100 0 0.049 0.068 −0.153 0.047 0.250
theta_hat_t_stats 100 0 0.655 0.941 −2.270 0.685 3.169
R2_demand 100 0 0.570 0.032 0.490 0.574 0.662
R2_supply 100 0 0.961 0.003 0.954 0.961 0.968
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.010 0.126 9.731 9.995 10.335
alpha1_hat 100 0 1.019 0.463 0.015 1.021 2.032
alpha2_hat 100 0 1.000 0.053 0.850 1.000 1.129
alpha3_hat 100 0 1.003 0.031 0.938 1.005 1.062
gamma0_hat 100 0 1.011 0.219 0.519 1.010 1.585
gamma1_hat 100 0 0.967 0.522 −0.138 0.965 1.991
gamma2_hat 100 0 1.001 0.032 0.916 1.003 1.066
gamma3_hat 100 0 0.996 0.029 0.912 0.993 1.082
theta_hat 100 0 0.050 0.050 −0.054 0.047 0.180
theta_hat_t_stats 100 0 0.979 0.986 −1.116 0.942 3.588
R2_demand 100 0 0.571 0.024 0.493 0.572 0.614
R2_supply 100 0 0.960 0.002 0.953 0.960 0.966
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.012 0.085 9.768 10.013 10.200
alpha1_hat 100 0 1.054 0.308 0.219 1.050 1.716
alpha2_hat 100 0 0.997 0.033 0.930 0.993 1.075
alpha3_hat 100 0 1.003 0.020 0.959 1.002 1.049
gamma0_hat 100 0 1.014 0.143 0.643 1.020 1.363
gamma1_hat 100 0 0.986 0.317 0.115 1.006 1.869
gamma2_hat 100 0 1.000 0.020 0.954 1.000 1.051
gamma3_hat 100 0 0.998 0.021 0.933 0.999 1.050
theta_hat 100 0 0.049 0.033 −0.035 0.047 0.129
theta_hat_t_stats 100 0 1.529 1.013 −1.084 1.499 3.956
R2_demand 100 0 0.567 0.017 0.533 0.569 0.607
R2_supply 100 0 0.960 0.002 0.956 0.960 0.963

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.027 0.973 8.184 9.967 14.140
alpha1_hat 100 0 2.793 15.115 −34.879 2.222 36.412
alpha2_hat 100 0 4.848 1.764 1.841 4.817 9.933
alpha3_hat 100 0 1.004 0.222 0.481 0.970 1.743
gamma0_hat 100 0 2.022 1.305 −3.630 2.208 4.270
gamma1_hat 100 0 3.204 14.739 −30.815 3.709 53.303
gamma2_hat 100 0 0.864 0.188 0.500 0.839 1.486
gamma3_hat 100 0 0.903 0.194 0.452 0.896 1.644
theta_hat 100 0 −0.100 0.340 −1.103 −0.086 0.803
theta_hat_t_stats 100 0 −0.452 1.004 −3.687 −0.325 1.673
R2_demand 100 0 0.591 0.189 −0.415 0.620 0.827
R2_supply 100 0 0.963 0.019 0.824 0.967 0.985
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.961 0.545 8.679 9.945 11.240
alpha1_hat 100 0 1.358 8.303 −18.125 1.695 18.536
alpha2_hat 100 0 4.939 0.965 2.676 4.963 7.482
alpha3_hat 100 0 0.994 0.134 0.628 0.996 1.392
gamma0_hat 100 0 1.428 0.810 −1.160 1.439 3.376
gamma1_hat 100 0 −1.460 9.161 −28.331 −1.189 20.042
gamma2_hat 100 0 0.956 0.122 0.655 0.951 1.311
gamma3_hat 100 0 0.960 0.140 0.639 0.980 1.343
theta_hat 100 0 0.051 0.203 −0.372 0.042 0.774
theta_hat_t_stats 100 0 0.162 0.961 −2.545 0.179 2.633
R2_demand 100 0 0.622 0.105 0.291 0.640 0.816
R2_supply 100 0 0.960 0.010 0.927 0.962 0.981
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.967 0.396 9.046 9.941 10.802
alpha1_hat 100 0 0.952 7.211 −15.549 1.057 18.070
alpha2_hat 100 0 4.972 0.807 3.013 4.915 6.846
alpha3_hat 100 0 1.001 0.099 0.769 0.993 1.269
gamma0_hat 100 0 1.267 0.670 −0.470 1.365 2.621
gamma1_hat 100 0 0.813 6.923 −15.620 0.314 22.408
gamma2_hat 100 0 0.967 0.100 0.728 0.962 1.316
gamma3_hat 100 0 0.977 0.103 0.741 0.967 1.235
theta_hat 100 0 0.027 0.164 −0.488 0.020 0.422
theta_hat_t_stats 100 0 0.071 1.086 −3.270 0.138 2.138
R2_demand 100 0 0.620 0.068 0.439 0.633 0.746
R2_supply 100 0 0.958 0.007 0.930 0.958 0.970
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.021 0.196 9.574 9.984 10.626
alpha1_hat 100 0 1.292 2.799 −5.567 0.953 9.036
alpha2_hat 100 0 4.988 0.341 4.140 4.967 5.810
alpha3_hat 100 0 1.001 0.047 0.903 1.002 1.120
gamma0_hat 100 0 1.069 0.291 0.176 1.051 1.704
gamma1_hat 100 0 1.034 2.758 −3.922 1.195 7.452
gamma2_hat 100 0 0.991 0.043 0.897 0.991 1.110
gamma3_hat 100 0 0.997 0.046 0.879 0.996 1.104
theta_hat 100 0 0.041 0.064 −0.109 0.037 0.168
theta_hat_t_stats 100 0 0.621 1.003 −1.787 0.572 2.568
R2_demand 100 0 0.617 0.036 0.502 0.619 0.691
R2_supply 100 0 0.956 0.003 0.947 0.955 0.962
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.139 9.664 9.999 10.391
alpha1_hat 100 0 0.892 1.788 −4.114 0.834 4.986
alpha2_hat 100 0 5.011 0.207 4.459 5.035 5.508
alpha3_hat 100 0 0.999 0.029 0.933 0.997 1.060
gamma0_hat 100 0 1.052 0.189 0.588 1.060 1.547
gamma1_hat 100 0 0.872 2.015 −4.808 0.786 5.908
gamma2_hat 100 0 0.996 0.028 0.925 0.997 1.071
gamma3_hat 100 0 0.994 0.032 0.924 0.992 1.067
theta_hat 100 0 0.046 0.044 −0.063 0.046 0.177
theta_hat_t_stats 100 0 1.006 0.960 −1.501 1.034 3.637
R2_demand 100 0 0.619 0.023 0.550 0.624 0.662
R2_supply 100 0 0.955 0.002 0.948 0.955 0.961
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.082 9.776 9.995 10.227
alpha1_hat 100 0 0.978 1.209 −2.241 1.031 4.140
alpha2_hat 100 0 5.003 0.142 4.679 4.974 5.388
alpha3_hat 100 0 1.001 0.019 0.959 1.000 1.047
gamma0_hat 100 0 1.024 0.119 0.750 1.019 1.307
gamma1_hat 100 0 0.782 1.232 −2.211 0.867 3.376
gamma2_hat 100 0 0.998 0.018 0.952 0.998 1.043
gamma3_hat 100 0 0.999 0.019 0.926 1.000 1.045
theta_hat 100 0 0.051 0.024 −0.003 0.049 0.113
theta_hat_t_stats 100 0 1.798 0.840 −0.113 1.686 3.852
R2_demand 100 0 0.620 0.014 0.587 0.621 0.653
R2_supply 100 0 0.955 0.001 0.950 0.955 0.958

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.078 1.156 7.657 10.040 17.632
alpha1_hat 100 0 2.394 47.415 −116.140 1.858 116.686
alpha2_hat 100 0 20.067 5.963 8.920 19.537 49.105
alpha3_hat 100 0 1.028 0.243 0.532 0.999 2.502
gamma0_hat 100 0 2.023 0.969 −0.202 2.004 4.375
gamma1_hat 100 0 10.932 47.612 −87.059 10.111 197.493
gamma2_hat 100 0 0.882 0.175 0.481 0.892 1.340
gamma3_hat 100 0 0.880 0.180 0.542 0.871 1.367
theta_hat 100 0 −0.122 0.290 −1.419 −0.092 0.491
theta_hat_t_stats 100 0 −0.467 1.009 −3.350 −0.309 1.613
R2_demand 100 0 0.564 0.537 −4.527 0.655 0.827
R2_supply 100 0 0.963 0.012 0.918 0.965 0.982
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.682 8.803 9.929 12.743
alpha1_hat 100 0 2.544 33.319 −74.863 0.205 79.604
alpha2_hat 100 0 19.850 3.830 11.814 20.057 30.398
alpha3_hat 100 0 0.992 0.157 0.692 0.984 1.611
gamma0_hat 100 0 1.474 0.754 −0.471 1.439 3.302
gamma1_hat 100 0 −2.563 34.680 −114.154 −0.063 84.861
gamma2_hat 100 0 0.958 0.122 0.630 0.954 1.214
gamma3_hat 100 0 0.945 0.124 0.652 0.937 1.295
theta_hat 100 0 0.010 0.185 −0.363 0.011 0.657
theta_hat_t_stats 100 0 −0.032 0.973 −2.504 0.041 2.455
R2_demand 100 0 0.618 0.129 −0.022 0.636 0.788
R2_supply 100 0 0.959 0.009 0.921 0.961 0.976
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.994 0.371 9.163 10.035 11.065
alpha1_hat 100 0 0.532 21.853 −67.392 −0.577 48.025
alpha2_hat 100 0 20.039 2.642 13.664 20.145 27.673
alpha3_hat 100 0 0.991 0.103 0.703 1.002 1.231
gamma0_hat 100 0 1.270 0.752 −0.605 1.341 2.805
gamma1_hat 100 0 3.551 19.623 −57.102 2.819 71.044
gamma2_hat 100 0 0.977 0.105 0.750 0.970 1.235
gamma3_hat 100 0 0.973 0.104 0.770 0.959 1.361
theta_hat 100 0 0.003 0.132 −0.288 0.009 0.349
theta_hat_t_stats 100 0 −0.054 0.943 −2.511 0.068 2.407
R2_demand 100 0 0.633 0.073 0.391 0.644 0.773
R2_supply 100 0 0.956 0.008 0.926 0.957 0.971
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.019 0.174 9.616 10.026 10.367
alpha1_hat 100 0 −0.831 10.322 −28.642 −1.953 26.857
alpha2_hat 100 0 20.251 1.177 16.694 20.211 23.084
alpha3_hat 100 0 1.006 0.041 0.912 1.012 1.103
gamma0_hat 100 0 1.016 0.296 0.263 1.015 1.925
gamma1_hat 100 0 −1.337 10.679 −27.553 −2.566 24.421
gamma2_hat 100 0 1.002 0.043 0.883 1.001 1.141
gamma3_hat 100 0 0.995 0.045 0.859 0.995 1.124
theta_hat 100 0 0.058 0.061 −0.081 0.060 0.181
theta_hat_t_stats 100 0 0.910 0.957 −1.476 0.978 2.914
R2_demand 100 0 0.634 0.028 0.545 0.635 0.696
R2_supply 100 0 0.953 0.004 0.940 0.953 0.962
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.999 0.137 9.610 10.001 10.327
alpha1_hat 100 0 1.356 8.204 −15.540 0.259 21.595
alpha2_hat 100 0 19.974 0.969 18.050 19.986 22.431
alpha3_hat 100 0 1.004 0.032 0.889 1.004 1.073
gamma0_hat 100 0 1.036 0.194 0.421 1.048 1.407
gamma1_hat 100 0 1.522 7.081 −16.821 0.746 19.011
gamma2_hat 100 0 0.997 0.031 0.929 0.997 1.090
gamma3_hat 100 0 0.994 0.033 0.910 0.990 1.074
theta_hat 100 0 0.043 0.042 −0.044 0.043 0.151
theta_hat_t_stats 100 0 0.954 0.927 −1.134 0.990 3.320
R2_demand 100 0 0.630 0.022 0.580 0.629 0.688
R2_supply 100 0 0.954 0.003 0.945 0.954 0.959
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.991 0.076 9.851 9.981 10.176
alpha1_hat 100 0 0.919 4.200 −9.902 0.690 14.379
alpha2_hat 100 0 19.977 0.462 18.729 20.022 21.168
alpha3_hat 100 0 1.000 0.017 0.952 1.000 1.040
gamma0_hat 100 0 1.019 0.131 0.640 1.024 1.324
gamma1_hat 100 0 1.024 4.583 −10.209 0.988 13.059
gamma2_hat 100 0 0.998 0.020 0.945 0.996 1.060
gamma3_hat 100 0 0.998 0.019 0.952 1.001 1.050
theta_hat 100 0 0.047 0.026 −0.037 0.048 0.110
theta_hat_t_stats 100 0 1.692 0.918 −1.379 1.763 3.901
R2_demand 100 0 0.631 0.013 0.602 0.633 0.659
R2_supply 100 0 0.953 0.001 0.950 0.953 0.957

\(\theta = 0.1\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.922 1.004 7.786 9.863 14.467
alpha1_hat 100 0 0.980 0.866 −0.802 0.867 3.118
alpha2_hat 100 0 0.097 0.084 −0.101 0.096 0.362
alpha3_hat 100 0 0.993 0.247 0.475 0.952 2.143
gamma0_hat 100 0 1.883 1.178 −1.690 2.044 4.430
gamma1_hat 100 0 −0.181 7.744 −60.074 0.634 11.685
gamma2_hat 100 0 0.897 0.192 0.382 0.857 1.453
gamma3_hat 100 0 0.905 0.185 0.415 0.899 1.538
theta_hat 100 0 0.555 4.037 −4.422 0.036 31.624
theta_hat_t_stats 100 0 0.010 1.056 −2.199 0.057 3.058
R2_demand 100 0 0.313 0.320 −1.342 0.399 0.705
R2_supply 100 0 0.980 0.005 0.963 0.981 0.991
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.982 0.793 8.536 9.913 14.488
alpha1_hat 100 0 0.983 0.706 −0.476 0.892 4.420
alpha2_hat 100 0 0.101 0.061 −0.055 0.101 0.364
alpha3_hat 100 0 0.998 0.183 0.695 0.974 2.060
gamma0_hat 100 0 1.513 0.837 −0.551 1.647 3.335
gamma1_hat 100 0 1.399 6.389 −18.737 0.978 54.252
gamma2_hat 100 0 0.941 0.123 0.661 0.937 1.291
gamma3_hat 100 0 0.944 0.145 0.570 0.926 1.324
theta_hat 100 0 0.007 2.173 −11.964 0.013 10.867
theta_hat_t_stats 100 0 0.126 1.022 −2.114 0.011 2.594
R2_demand 100 0 0.279 0.311 −2.128 0.345 0.582
R2_supply 100 0 0.978 0.005 0.960 0.979 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.017 0.514 8.751 9.988 11.461
alpha1_hat 100 0 1.064 0.383 0.235 1.066 1.977
alpha2_hat 100 0 0.094 0.040 0.005 0.091 0.220
alpha3_hat 100 0 1.007 0.101 0.823 0.997 1.310
gamma0_hat 100 0 1.149 0.738 −0.773 1.159 2.612
gamma1_hat 100 0 0.835 1.475 −8.442 0.943 5.889
gamma2_hat 100 0 0.981 0.107 0.779 0.977 1.282
gamma3_hat 100 0 0.991 0.109 0.764 0.986 1.368
theta_hat 100 0 0.156 0.762 −2.439 0.102 4.720
theta_hat_t_stats 100 0 0.198 1.069 −2.378 0.289 2.668
R2_demand 100 0 0.310 0.134 −0.144 0.327 0.574
R2_supply 100 0 0.976 0.004 0.962 0.976 0.984
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.185 9.641 10.009 10.470
alpha1_hat 100 0 0.987 0.168 0.606 0.985 1.521
alpha2_hat 100 0 0.102 0.016 0.055 0.102 0.131
alpha3_hat 100 0 0.998 0.046 0.873 0.998 1.120
gamma0_hat 100 0 1.047 0.315 0.199 1.047 1.819
gamma1_hat 100 0 1.030 0.314 0.193 1.007 1.784
gamma2_hat 100 0 0.993 0.043 0.892 0.995 1.084
gamma3_hat 100 0 0.996 0.050 0.879 0.999 1.103
theta_hat 100 0 0.077 0.167 −0.328 0.084 0.564
theta_hat_t_stats 100 0 0.443 1.025 −2.221 0.483 3.478
R2_demand 100 0 0.313 0.051 0.186 0.323 0.403
R2_supply 100 0 0.976 0.002 0.971 0.976 0.980
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.150 9.650 10.002 10.424
alpha1_hat 100 0 0.987 0.123 0.569 0.993 1.312
alpha2_hat 100 0 0.101 0.012 0.072 0.101 0.125
alpha3_hat 100 0 1.001 0.036 0.910 1.001 1.110
gamma0_hat 100 0 0.995 0.223 0.377 0.982 1.509
gamma1_hat 100 0 0.983 0.244 0.157 0.991 1.477
gamma2_hat 100 0 1.001 0.033 0.919 0.996 1.101
gamma3_hat 100 0 1.001 0.032 0.918 1.005 1.079
theta_hat 100 0 0.110 0.126 −0.154 0.102 0.506
theta_hat_t_stats 100 0 0.913 1.021 −1.245 0.921 3.622
R2_demand 100 0 0.312 0.040 0.198 0.314 0.415
R2_supply 100 0 0.975 0.001 0.973 0.975 0.979
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.082 9.812 10.004 10.206
alpha1_hat 100 0 0.989 0.081 0.798 0.978 1.193
alpha2_hat 100 0 0.102 0.008 0.084 0.101 0.120
alpha3_hat 100 0 1.002 0.020 0.962 1.003 1.054
gamma0_hat 100 0 0.984 0.133 0.566 0.984 1.291
gamma1_hat 100 0 1.021 0.130 0.611 1.025 1.446
gamma2_hat 100 0 1.003 0.020 0.947 1.000 1.050
gamma3_hat 100 0 1.001 0.019 0.947 1.002 1.050
theta_hat 100 0 0.092 0.066 −0.093 0.091 0.295
theta_hat_t_stats 100 0 1.251 0.891 −1.131 1.246 3.730
R2_demand 100 0 0.312 0.026 0.243 0.314 0.360
R2_supply 100 0 0.975 0.001 0.973 0.975 0.978

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.978 1.100 7.673 9.744 14.551
alpha1_hat 100 0 1.068 2.043 −3.604 1.104 9.409
alpha2_hat 100 0 0.491 0.222 −0.065 0.470 1.043
alpha3_hat 100 0 0.979 0.220 0.433 0.949 1.703
gamma0_hat 100 0 1.900 1.339 −2.227 1.957 5.137
gamma1_hat 100 0 29.896 293.294 −26.760 1.114 2933.281
gamma2_hat 100 0 0.901 0.198 0.390 0.902 1.466
gamma3_hat 100 0 0.903 0.198 0.382 0.879 1.365
theta_hat 100 0 −6.465 65.218 −652.088 −0.084 5.690
theta_hat_t_stats 100 0 −0.139 0.947 −2.409 −0.304 2.774
R2_demand 100 0 0.441 0.308 −1.486 0.501 0.798
R2_supply 100 0 0.973 0.009 0.940 0.975 0.988
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.938 0.726 8.691 9.844 12.232
alpha1_hat 100 0 1.088 1.283 −2.326 1.066 5.789
alpha2_hat 100 0 0.484 0.140 0.028 0.491 0.981
alpha3_hat 100 0 0.991 0.157 0.605 0.962 1.598
gamma0_hat 100 0 1.636 0.863 −0.486 1.676 3.808
gamma1_hat 100 0 0.467 4.925 −45.816 0.914 6.039
gamma2_hat 100 0 0.919 0.144 0.602 0.902 1.431
gamma3_hat 100 0 0.921 0.130 0.584 0.917 1.216
theta_hat 100 0 0.128 1.024 −0.765 0.017 9.788
theta_hat_t_stats 100 0 0.030 1.054 −3.030 0.076 2.657
R2_demand 100 0 0.469 0.192 −0.342 0.511 0.774
R2_supply 100 0 0.971 0.006 0.952 0.972 0.984
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.041 0.475 9.076 10.000 11.728
alpha1_hat 100 0 1.054 1.037 −1.468 0.926 3.057
alpha2_hat 100 0 0.500 0.112 0.204 0.497 0.780
alpha3_hat 100 0 1.010 0.110 0.765 1.016 1.246
gamma0_hat 100 0 1.149 0.731 −1.035 1.233 2.715
gamma1_hat 100 0 1.031 1.049 −1.254 1.007 4.229
gamma2_hat 100 0 0.988 0.113 0.763 0.983 1.256
gamma3_hat 100 0 0.963 0.096 0.758 0.949 1.288
theta_hat 100 0 0.070 0.205 −0.377 0.058 0.602
theta_hat_t_stats 100 0 0.299 1.045 −2.918 0.301 2.686
R2_demand 100 0 0.477 0.126 −0.315 0.500 0.643
R2_supply 100 0 0.968 0.006 0.949 0.968 0.978
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.042 0.185 9.652 10.034 10.520
alpha1_hat 100 0 0.969 0.399 −0.250 0.997 2.022
alpha2_hat 100 0 0.509 0.044 0.410 0.505 0.621
alpha3_hat 100 0 1.014 0.047 0.912 1.011 1.141
gamma0_hat 100 0 1.074 0.252 0.481 1.083 1.657
gamma1_hat 100 0 0.934 0.398 −0.084 0.935 1.819
gamma2_hat 100 0 0.990 0.041 0.909 0.987 1.098
gamma3_hat 100 0 0.996 0.040 0.915 0.996 1.107
theta_hat 100 0 0.101 0.069 −0.026 0.097 0.281
theta_hat_t_stats 100 0 1.220 0.820 −0.358 1.179 3.228
R2_demand 100 0 0.480 0.045 0.378 0.482 0.572
R2_supply 100 0 0.967 0.002 0.961 0.967 0.972
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.972 0.131 9.681 9.966 10.273
alpha1_hat 100 0 0.966 0.276 −0.048 0.946 1.588
alpha2_hat 100 0 0.500 0.029 0.443 0.499 0.601
alpha3_hat 100 0 0.997 0.031 0.923 0.993 1.096
gamma0_hat 100 0 1.037 0.206 0.433 1.043 1.536
gamma1_hat 100 0 0.968 0.344 0.090 0.958 1.711
gamma2_hat 100 0 0.996 0.028 0.930 0.995 1.077
gamma3_hat 100 0 0.997 0.030 0.937 0.995 1.079
theta_hat 100 0 0.101 0.061 −0.025 0.093 0.280
theta_hat_t_stats 100 0 1.696 1.007 −0.489 1.557 4.285
R2_demand 100 0 0.494 0.031 0.424 0.492 0.594
R2_supply 100 0 0.967 0.002 0.961 0.967 0.971
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.005 0.087 9.848 9.996 10.264
alpha1_hat 100 0 1.028 0.182 0.705 1.010 1.504
alpha2_hat 100 0 0.498 0.019 0.447 0.501 0.534
alpha3_hat 100 0 0.999 0.019 0.961 0.995 1.057
gamma0_hat 100 0 1.043 0.121 0.704 1.045 1.306
gamma1_hat 100 0 1.002 0.224 0.414 1.000 1.450
gamma2_hat 100 0 0.995 0.019 0.956 0.997 1.054
gamma3_hat 100 0 0.996 0.019 0.951 0.996 1.045
theta_hat 100 0 0.094 0.039 0.020 0.096 0.205
theta_hat_t_stats 100 0 2.496 0.995 0.579 2.550 5.253
R2_demand 100 0 0.490 0.020 0.428 0.492 0.528
R2_supply 100 0 0.967 0.001 0.964 0.966 0.970

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.038 0.939 8.563 9.859 13.464
alpha1_hat 100 0 1.139 3.316 −7.905 0.999 9.244
alpha2_hat 100 0 0.998 0.364 0.259 0.969 1.958
alpha3_hat 100 0 1.000 0.203 0.591 0.988 1.603
gamma0_hat 100 0 1.684 1.154 −1.939 1.773 3.943
gamma1_hat 100 0 1.637 4.333 −14.086 1.848 21.871
gamma2_hat 100 0 0.925 0.183 0.490 0.921 1.405
gamma3_hat 100 0 0.926 0.199 0.295 0.932 1.581
theta_hat 100 0 −0.030 0.423 −1.460 −0.060 1.663
theta_hat_t_stats 100 0 −0.239 0.995 −3.483 −0.182 2.359
R2_demand 100 0 0.501 0.210 −0.175 0.550 0.766
R2_supply 100 0 0.970 0.009 0.936 0.972 0.983
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.699 8.589 10.008 12.681
alpha1_hat 100 0 0.966 2.437 −7.025 0.897 6.588
alpha2_hat 100 0 1.004 0.283 0.358 1.009 1.796
alpha3_hat 100 0 0.992 0.162 0.666 0.982 1.479
gamma0_hat 100 0 1.385 0.896 −1.127 1.428 3.690
gamma1_hat 100 0 0.830 3.255 −11.462 1.248 8.546
gamma2_hat 100 0 0.962 0.141 0.534 0.973 1.317
gamma3_hat 100 0 0.952 0.122 0.689 0.963 1.272
theta_hat 100 0 0.067 0.352 −0.563 0.018 1.686
theta_hat_t_stats 100 0 0.105 1.116 −2.763 0.087 2.506
R2_demand 100 0 0.535 0.142 0.009 0.559 0.772
R2_supply 100 0 0.966 0.008 0.940 0.967 0.981
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.052 0.470 9.294 9.975 11.816
alpha1_hat 100 0 0.985 1.597 −3.600 1.158 5.063
alpha2_hat 100 0 1.013 0.187 0.547 0.993 1.675
alpha3_hat 100 0 1.009 0.109 0.786 1.004 1.354
gamma0_hat 100 0 1.232 0.547 −0.277 1.310 2.631
gamma1_hat 100 0 0.948 1.763 −3.234 0.880 5.748
gamma2_hat 100 0 0.976 0.087 0.781 0.965 1.203
gamma3_hat 100 0 0.972 0.088 0.755 0.976 1.251
theta_hat 100 0 0.071 0.175 −0.340 0.070 0.610
theta_hat_t_stats 100 0 0.397 0.988 −1.829 0.417 2.747
R2_demand 100 0 0.529 0.112 −0.065 0.552 0.714
R2_supply 100 0 0.965 0.006 0.943 0.966 0.976
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.016 0.198 9.512 9.995 10.631
alpha1_hat 100 0 1.035 0.685 −1.053 1.009 3.025
alpha2_hat 100 0 0.999 0.082 0.797 1.003 1.248
alpha3_hat 100 0 0.997 0.049 0.896 0.992 1.156
gamma0_hat 100 0 1.017 0.344 0.169 1.015 1.902
gamma1_hat 100 0 1.061 0.768 −0.559 1.026 2.888
gamma2_hat 100 0 0.998 0.050 0.876 1.005 1.115
gamma3_hat 100 0 0.991 0.048 0.863 0.990 1.137
theta_hat 100 0 0.093 0.082 −0.104 0.101 0.251
theta_hat_t_stats 100 0 1.210 1.072 −1.469 1.325 2.950
R2_demand 100 0 0.529 0.041 0.420 0.531 0.624
R2_supply 100 0 0.963 0.003 0.955 0.963 0.969
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.994 0.127 9.689 9.981 10.334
alpha1_hat 100 0 1.076 0.462 0.143 1.053 2.293
alpha2_hat 100 0 0.991 0.046 0.883 0.987 1.095
alpha3_hat 100 0 1.002 0.031 0.902 1.001 1.075
gamma0_hat 100 0 1.041 0.205 0.502 1.043 1.514
gamma1_hat 100 0 0.983 0.481 −0.670 1.017 2.157
gamma2_hat 100 0 0.996 0.029 0.920 0.997 1.080
gamma3_hat 100 0 0.993 0.033 0.922 0.991 1.065
theta_hat 100 0 0.095 0.049 −0.038 0.091 0.266
theta_hat_t_stats 100 0 1.797 0.903 −0.742 1.726 4.567
R2_demand 100 0 0.540 0.023 0.480 0.543 0.612
R2_supply 100 0 0.963 0.002 0.957 0.963 0.968
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.002 0.094 9.754 10.008 10.219
alpha1_hat 100 0 0.994 0.269 0.298 0.992 1.531
alpha2_hat 100 0 1.001 0.032 0.908 0.999 1.076
alpha3_hat 100 0 1.000 0.021 0.955 0.997 1.054
gamma0_hat 100 0 1.021 0.116 0.744 1.022 1.315
gamma1_hat 100 0 1.002 0.267 0.372 0.981 1.529
gamma2_hat 100 0 0.997 0.019 0.951 0.997 1.038
gamma3_hat 100 0 0.995 0.020 0.949 0.998 1.037
theta_hat 100 0 0.098 0.027 0.034 0.099 0.163
theta_hat_t_stats 100 0 2.950 0.783 1.018 3.001 4.782
R2_demand 100 0 0.539 0.018 0.494 0.538 0.586
R2_supply 100 0 0.963 0.001 0.959 0.963 0.966

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.972 0.721 8.034 9.826 12.227
alpha1_hat 100 0 0.732 12.626 −30.189 1.279 31.383
alpha2_hat 100 0 5.012 1.345 2.416 4.866 8.509
alpha3_hat 100 0 1.007 0.213 0.575 0.995 1.646
gamma0_hat 100 0 1.895 1.125 −0.508 1.919 4.721
gamma1_hat 100 0 0.277 11.973 −34.682 0.914 31.482
gamma2_hat 100 0 0.899 0.185 0.550 0.880 1.428
gamma3_hat 100 0 0.900 0.184 0.528 0.875 1.461
theta_hat 100 0 −0.005 0.250 −0.505 −0.017 0.620
theta_hat_t_stats 100 0 −0.104 0.888 −2.407 −0.053 1.741
R2_demand 100 0 0.563 0.203 −0.275 0.602 0.819
R2_supply 100 0 0.968 0.009 0.942 0.969 0.985
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.938 0.567 8.736 9.990 11.303
alpha1_hat 100 0 1.609 10.753 −29.624 1.950 38.203
alpha2_hat 100 0 4.877 1.207 1.285 4.847 9.029
alpha3_hat 100 0 1.003 0.133 0.770 0.995 1.363
gamma0_hat 100 0 1.394 0.791 −0.751 1.496 3.673
gamma1_hat 100 0 −1.047 10.376 −52.175 −1.105 27.140
gamma2_hat 100 0 0.961 0.122 0.678 0.945 1.312
gamma3_hat 100 0 0.973 0.140 0.618 0.972 1.358
theta_hat 100 0 0.092 0.230 −0.486 0.069 1.263
theta_hat_t_stats 100 0 0.318 0.806 −1.907 0.343 2.151
R2_demand 100 0 0.591 0.112 0.243 0.613 0.799
R2_supply 100 0 0.963 0.008 0.938 0.965 0.978
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.030 0.447 8.912 9.959 11.050
alpha1_hat 100 0 0.800 6.976 −16.810 0.746 18.238
alpha2_hat 100 0 5.048 0.844 3.433 5.046 7.572
alpha3_hat 100 0 1.011 0.106 0.760 1.009 1.282
gamma0_hat 100 0 1.176 0.613 −0.374 1.209 2.581
gamma1_hat 100 0 1.392 6.874 −16.177 0.979 20.647
gamma2_hat 100 0 0.984 0.095 0.798 0.985 1.193
gamma3_hat 100 0 0.985 0.099 0.774 0.973 1.220
theta_hat 100 0 0.070 0.153 −0.309 0.070 0.542
theta_hat_t_stats 100 0 0.422 0.949 −2.047 0.499 2.865
R2_demand 100 0 0.576 0.088 0.266 0.592 0.736
R2_supply 100 0 0.960 0.006 0.944 0.962 0.974
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.183 9.653 9.977 10.385
alpha1_hat 100 0 1.291 2.956 −5.136 1.479 7.205
alpha2_hat 100 0 4.973 0.360 4.186 4.986 5.969
alpha3_hat 100 0 1.008 0.041 0.921 1.010 1.142
gamma0_hat 100 0 0.991 0.294 0.157 0.996 1.625
gamma1_hat 100 0 1.091 2.829 −5.921 1.445 6.178
gamma2_hat 100 0 1.002 0.046 0.880 0.998 1.107
gamma3_hat 100 0 1.002 0.040 0.912 1.003 1.105
theta_hat 100 0 0.099 0.066 −0.063 0.100 0.277
theta_hat_t_stats 100 0 1.443 0.921 −1.087 1.460 3.610
R2_demand 100 0 0.588 0.034 0.509 0.589 0.674
R2_supply 100 0 0.958 0.003 0.944 0.959 0.965
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.985 0.133 9.706 9.966 10.504
alpha1_hat 100 0 0.858 2.075 −4.874 0.888 6.281
alpha2_hat 100 0 4.998 0.252 4.389 4.956 5.689
alpha3_hat 100 0 0.998 0.028 0.949 0.999 1.086
gamma0_hat 100 0 1.030 0.210 0.420 1.030 1.653
gamma1_hat 100 0 0.729 2.014 −3.546 0.702 6.249
gamma2_hat 100 0 0.996 0.030 0.922 0.998 1.075
gamma3_hat 100 0 1.000 0.037 0.909 1.001 1.092
theta_hat 100 0 0.102 0.051 −0.023 0.105 0.224
theta_hat_t_stats 100 0 2.145 1.029 −0.518 2.196 4.506
R2_demand 100 0 0.589 0.025 0.499 0.591 0.629
R2_supply 100 0 0.958 0.002 0.953 0.959 0.964
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.999 0.086 9.818 9.988 10.300
alpha1_hat 100 0 0.973 1.200 −1.703 0.997 3.804
alpha2_hat 100 0 5.003 0.145 4.683 5.007 5.338
alpha3_hat 100 0 1.001 0.020 0.954 0.999 1.047
gamma0_hat 100 0 1.025 0.134 0.759 1.026 1.324
gamma1_hat 100 0 1.152 1.316 −1.468 1.181 4.099
gamma2_hat 100 0 0.998 0.020 0.956 0.996 1.045
gamma3_hat 100 0 0.996 0.018 0.944 0.994 1.046
theta_hat 100 0 0.094 0.029 0.024 0.097 0.169
theta_hat_t_stats 100 0 3.153 0.949 0.825 3.228 5.524
R2_demand 100 0 0.586 0.019 0.531 0.586 0.626
R2_supply 100 0 0.958 0.001 0.955 0.959 0.962

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.009 0.849 8.005 9.985 12.852
alpha1_hat 100 0 −1.334 57.303 −166.001 −4.268 170.300
alpha2_hat 100 0 20.221 6.912 0.669 19.854 41.702
alpha3_hat 100 0 0.983 0.197 0.505 0.978 1.448
gamma0_hat 100 0 1.902 1.153 −2.150 2.065 4.920
gamma1_hat 100 0 −15.038 93.190 −771.027 −1.985 110.467
gamma2_hat 100 0 0.910 0.180 0.337 0.911 1.361
gamma3_hat 100 0 0.915 0.186 0.556 0.911 1.759
theta_hat 100 0 0.077 0.548 −0.738 0.001 4.270
theta_hat_t_stats 100 0 −0.035 1.017 −2.660 0.002 2.637
R2_demand 100 0 0.585 0.161 −0.060 0.619 0.858
R2_supply 100 0 0.968 0.011 0.916 0.971 0.984
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.136 0.644 8.713 10.140 11.647
alpha1_hat 100 0 −0.463 41.128 −102.521 0.213 130.323
alpha2_hat 100 0 20.640 4.822 8.791 20.220 31.189
alpha3_hat 100 0 1.009 0.145 0.699 1.004 1.313
gamma0_hat 100 0 1.389 0.842 −1.109 1.389 3.649
gamma1_hat 100 0 2.422 38.098 −90.010 6.290 94.297
gamma2_hat 100 0 0.963 0.146 0.634 0.963 1.390
gamma3_hat 100 0 0.946 0.133 0.672 0.932 1.314
theta_hat 100 0 0.036 0.203 −0.361 0.027 0.544
theta_hat_t_stats 100 0 0.116 1.002 −2.375 0.125 2.473
R2_demand 100 0 0.563 0.146 0.035 0.586 0.793
R2_supply 100 0 0.963 0.007 0.934 0.963 0.977
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.963 0.437 9.013 9.902 11.204
alpha1_hat 100 0 −3.256 26.239 −73.104 −0.447 61.611
alpha2_hat 100 0 20.285 3.231 13.759 20.306 30.019
alpha3_hat 100 0 0.998 0.111 0.765 0.996 1.329
gamma0_hat 100 0 1.186 0.637 −0.332 1.246 2.400
gamma1_hat 100 0 0.205 22.736 −72.782 −2.433 79.513
gamma2_hat 100 0 0.980 0.092 0.718 0.974 1.161
gamma3_hat 100 0 0.987 0.101 0.763 0.982 1.247
theta_hat 100 0 0.083 0.139 −0.343 0.085 0.446
theta_hat_t_stats 100 0 0.519 0.940 −2.401 0.583 2.429
R2_demand 100 0 0.594 0.078 0.364 0.594 0.772
R2_supply 100 0 0.960 0.006 0.938 0.960 0.973
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.995 0.207 9.460 10.015 10.476
alpha1_hat 100 0 0.251 10.457 −23.319 2.316 21.523
alpha2_hat 100 0 20.076 1.288 16.737 19.948 23.231
alpha3_hat 100 0 0.999 0.049 0.851 0.997 1.119
gamma0_hat 100 0 0.977 0.344 0.095 0.934 1.627
gamma1_hat 100 0 1.228 10.007 −18.116 1.840 30.003
gamma2_hat 100 0 1.005 0.049 0.910 0.997 1.134
gamma3_hat 100 0 1.007 0.042 0.898 1.007 1.121
theta_hat 100 0 0.100 0.060 −0.036 0.109 0.217
theta_hat_t_stats 100 0 1.503 0.878 −0.596 1.702 3.071
R2_demand 100 0 0.595 0.039 0.461 0.595 0.667
R2_supply 100 0 0.957 0.003 0.948 0.957 0.966
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.142 9.640 9.988 10.326
alpha1_hat 100 0 0.453 8.219 −19.675 0.204 15.929
alpha2_hat 100 0 20.017 0.956 17.485 20.057 22.606
alpha3_hat 100 0 0.995 0.034 0.903 0.992 1.085
gamma0_hat 100 0 1.046 0.209 0.560 1.062 1.468
gamma1_hat 100 0 0.898 8.416 −18.467 −0.328 23.787
gamma2_hat 100 0 0.990 0.029 0.920 0.990 1.068
gamma3_hat 100 0 0.999 0.031 0.935 0.998 1.084
theta_hat 100 0 0.098 0.049 −0.034 0.099 0.204
theta_hat_t_stats 100 0 2.100 0.987 −0.756 2.167 4.293
R2_demand 100 0 0.597 0.026 0.518 0.600 0.642
R2_supply 100 0 0.957 0.002 0.953 0.958 0.963
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.091 9.785 9.992 10.190
alpha1_hat 100 0 0.643 4.690 −10.223 0.769 11.030
alpha2_hat 100 0 20.054 0.586 18.926 20.051 21.517
alpha3_hat 100 0 1.001 0.020 0.945 1.003 1.045
gamma0_hat 100 0 1.033 0.145 0.710 1.033 1.372
gamma1_hat 100 0 2.303 4.683 −8.809 2.237 19.061
gamma2_hat 100 0 0.995 0.022 0.952 0.996 1.052
gamma3_hat 100 0 0.998 0.020 0.951 0.997 1.056
theta_hat 100 0 0.090 0.029 0.006 0.094 0.151
theta_hat_t_stats 100 0 3.111 0.950 0.192 3.231 5.120
R2_demand 100 0 0.594 0.017 0.556 0.596 0.628
R2_supply 100 0 0.957 0.002 0.953 0.957 0.960

\(\theta = 0.2\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.075 1.011 7.398 10.061 13.437
alpha1_hat 100 0 1.091 0.872 −0.569 1.004 5.993
alpha2_hat 100 0 0.095 0.088 −0.270 0.097 0.283
alpha3_hat 100 0 1.004 0.238 0.588 0.981 1.694
gamma0_hat 100 0 1.922 1.327 −3.146 2.043 5.539
gamma1_hat 100 0 2.018 16.662 −60.991 0.797 119.484
gamma2_hat 100 0 0.897 0.195 0.465 0.886 1.393
gamma3_hat 100 0 0.879 0.181 0.400 0.874 1.276
theta_hat 100 0 −0.612 9.660 −76.883 0.085 33.313
theta_hat_t_stats 100 0 0.117 1.032 −2.066 0.121 2.575
R2_demand 100 0 0.229 0.389 −1.627 0.339 0.609
R2_supply 100 0 0.982 0.005 0.962 0.982 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.926 0.661 8.653 9.862 11.727
alpha1_hat 100 0 0.794 0.635 −0.550 0.785 2.855
alpha2_hat 100 0 0.117 0.064 −0.036 0.113 0.357
alpha3_hat 100 0 1.000 0.145 0.676 0.988 1.460
gamma0_hat 100 0 1.334 0.865 −0.986 1.396 3.549
gamma1_hat 100 0 0.875 1.753 −6.103 0.900 10.423
gamma2_hat 100 0 0.965 0.132 0.691 0.959 1.238
gamma3_hat 100 0 0.976 0.133 0.586 0.991 1.407
theta_hat 100 0 0.192 0.942 −5.125 0.153 3.781
theta_hat_t_stats 100 0 0.300 0.939 −1.698 0.333 2.366
R2_demand 100 0 0.269 0.185 −0.489 0.302 0.527
R2_supply 100 0 0.979 0.004 0.962 0.980 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.959 0.434 8.882 9.918 11.708
alpha1_hat 100 0 0.941 0.458 −0.074 0.949 2.347
alpha2_hat 100 0 0.104 0.043 −0.022 0.102 0.237
alpha3_hat 100 0 1.001 0.102 0.755 0.998 1.358
gamma0_hat 100 0 1.228 0.672 −0.616 1.232 2.698
gamma1_hat 100 0 0.919 1.190 −1.929 0.952 7.253
gamma2_hat 100 0 0.973 0.104 0.707 0.963 1.232
gamma3_hat 100 0 0.976 0.100 0.722 0.978 1.212
theta_hat 100 0 0.221 0.637 −2.849 0.163 2.237
theta_hat_t_stats 100 0 0.464 1.015 −2.902 0.534 2.706
R2_demand 100 0 0.264 0.137 −0.149 0.274 0.532
R2_supply 100 0 0.978 0.004 0.967 0.979 0.986
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.031 0.176 9.642 10.023 10.461
alpha1_hat 100 0 1.010 0.190 0.643 1.021 1.701
alpha2_hat 100 0 0.101 0.019 0.043 0.098 0.152
alpha3_hat 100 0 1.008 0.037 0.932 1.008 1.111
gamma0_hat 100 0 1.077 0.336 0.256 1.071 2.095
gamma1_hat 100 0 0.975 0.363 −0.048 0.997 1.749
gamma2_hat 100 0 0.988 0.051 0.844 0.986 1.102
gamma3_hat 100 0 0.989 0.045 0.876 0.989 1.111
theta_hat 100 0 0.200 0.186 −0.293 0.191 0.696
theta_hat_t_stats 100 0 1.061 0.932 −1.529 1.097 3.074
R2_demand 100 0 0.264 0.056 0.134 0.271 0.373
R2_supply 100 0 0.977 0.002 0.973 0.977 0.981
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.132 9.671 10.024 10.394
alpha1_hat 100 0 1.011 0.132 0.646 1.015 1.329
alpha2_hat 100 0 0.099 0.012 0.069 0.098 0.137
alpha3_hat 100 0 0.999 0.030 0.918 1.001 1.077
gamma0_hat 100 0 0.981 0.207 0.464 0.998 1.505
gamma1_hat 100 0 0.963 0.242 0.483 0.939 1.742
gamma2_hat 100 0 1.002 0.030 0.927 1.001 1.065
gamma3_hat 100 0 1.002 0.031 0.921 0.999 1.064
theta_hat 100 0 0.223 0.129 −0.162 0.224 0.486
theta_hat_t_stats 100 0 1.721 0.961 −1.461 1.801 3.688
R2_demand 100 0 0.276 0.042 0.161 0.276 0.375
R2_supply 100 0 0.977 0.001 0.973 0.977 0.979
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.995 0.092 9.780 10.005 10.230
alpha1_hat 100 0 0.988 0.087 0.790 0.990 1.212
alpha2_hat 100 0 0.101 0.008 0.081 0.101 0.120
alpha3_hat 100 0 0.999 0.021 0.942 0.998 1.052
gamma0_hat 100 0 0.999 0.114 0.722 0.999 1.257
gamma1_hat 100 0 1.001 0.146 0.637 1.005 1.295
gamma2_hat 100 0 1.002 0.018 0.961 1.002 1.046
gamma3_hat 100 0 1.000 0.017 0.956 0.999 1.041
theta_hat 100 0 0.200 0.069 0.053 0.194 0.383
theta_hat_t_stats 100 0 2.542 0.860 0.757 2.508 4.948
R2_demand 100 0 0.279 0.026 0.191 0.279 0.337
R2_supply 100 0 0.977 0.001 0.975 0.977 0.979

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.931 0.852 7.610 9.913 12.363
alpha1_hat 100 0 0.945 2.071 −5.565 0.842 6.752
alpha2_hat 100 0 0.494 0.218 −0.055 0.491 1.175
alpha3_hat 100 0 0.991 0.212 0.508 0.965 1.470
gamma0_hat 100 0 1.797 1.196 −1.527 1.859 4.281
gamma1_hat 100 0 0.014 4.618 −35.087 0.584 5.702
gamma2_hat 100 0 0.897 0.177 0.507 0.892 1.283
gamma3_hat 100 0 0.892 0.174 0.464 0.898 1.458
theta_hat 100 0 0.265 0.789 −0.694 0.126 5.516
theta_hat_t_stats 100 0 0.297 0.967 −2.328 0.365 2.256
R2_demand 100 0 0.366 0.272 −0.637 0.422 0.733
R2_supply 100 0 0.978 0.007 0.953 0.979 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.609 8.668 9.966 11.644
alpha1_hat 100 0 0.873 1.411 −2.676 0.841 4.144
alpha2_hat 100 0 0.514 0.145 0.207 0.512 1.042
alpha3_hat 100 0 1.013 0.139 0.662 1.010 1.360
gamma0_hat 100 0 1.316 1.081 −2.379 1.490 3.081
gamma1_hat 100 0 0.779 1.677 −4.047 0.780 5.568
gamma2_hat 100 0 0.968 0.159 0.720 0.960 1.490
gamma3_hat 100 0 0.961 0.161 0.633 0.936 1.499
theta_hat 100 0 0.190 0.344 −0.459 0.143 1.123
theta_hat_t_stats 100 0 0.570 0.986 −1.700 0.491 3.177
R2_demand 100 0 0.411 0.176 −0.371 0.446 0.713
R2_supply 100 0 0.973 0.007 0.936 0.974 0.986
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.971 0.392 9.083 9.932 11.142
alpha1_hat 100 0 0.983 1.008 −1.916 1.021 3.446
alpha2_hat 100 0 0.498 0.106 0.255 0.489 0.800
alpha3_hat 100 0 0.983 0.095 0.814 0.981 1.286
gamma0_hat 100 0 1.035 0.655 −1.060 1.012 2.747
gamma1_hat 100 0 0.803 1.298 −2.974 0.898 4.067
gamma2_hat 100 0 0.997 0.091 0.762 0.989 1.262
gamma3_hat 100 0 1.000 0.092 0.795 0.993 1.304
theta_hat 100 0 0.231 0.231 −0.286 0.206 1.039
theta_hat_t_stats 100 0 1.005 0.957 −1.802 0.978 3.117
R2_demand 100 0 0.426 0.108 0.137 0.451 0.643
R2_supply 100 0 0.971 0.005 0.956 0.971 0.981
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.028 0.204 9.613 10.024 10.608
alpha1_hat 100 0 1.011 0.439 0.159 1.018 2.029
alpha2_hat 100 0 0.503 0.053 0.384 0.510 0.622
alpha3_hat 100 0 1.010 0.046 0.915 1.012 1.112
gamma0_hat 100 0 1.025 0.307 0.343 1.040 2.037
gamma1_hat 100 0 0.934 0.572 −0.361 0.928 2.846
gamma2_hat 100 0 0.996 0.046 0.860 0.998 1.119
gamma3_hat 100 0 0.997 0.042 0.882 1.000 1.111
theta_hat 100 0 0.207 0.108 −0.165 0.191 0.481
theta_hat_t_stats 100 0 2.221 1.054 −1.820 2.184 5.357
R2_demand 100 0 0.425 0.045 0.296 0.425 0.511
R2_supply 100 0 0.970 0.003 0.961 0.970 0.977
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.135 9.705 9.999 10.503
alpha1_hat 100 0 1.015 0.304 0.296 1.053 1.931
alpha2_hat 100 0 0.499 0.031 0.434 0.495 0.592
alpha3_hat 100 0 1.002 0.030 0.923 1.002 1.124
gamma0_hat 100 0 1.018 0.212 0.520 1.035 1.552
gamma1_hat 100 0 0.940 0.364 0.171 0.978 1.830
gamma2_hat 100 0 0.997 0.030 0.928 0.996 1.065
gamma3_hat 100 0 0.998 0.030 0.903 0.999 1.071
theta_hat 100 0 0.208 0.064 0.024 0.217 0.315
theta_hat_t_stats 100 0 3.214 0.956 0.401 3.314 4.738
R2_demand 100 0 0.429 0.035 0.313 0.426 0.516
R2_supply 100 0 0.970 0.002 0.965 0.970 0.974
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.086 9.814 9.999 10.170
alpha1_hat 100 0 1.027 0.208 0.573 1.011 1.535
alpha2_hat 100 0 0.498 0.024 0.444 0.498 0.558
alpha3_hat 100 0 1.001 0.021 0.959 0.999 1.056
gamma0_hat 100 0 1.018 0.145 0.611 1.018 1.417
gamma1_hat 100 0 0.974 0.241 0.317 0.985 1.641
gamma2_hat 100 0 0.998 0.021 0.952 0.998 1.058
gamma3_hat 100 0 0.996 0.022 0.940 0.997 1.060
theta_hat 100 0 0.202 0.040 0.111 0.202 0.304
theta_hat_t_stats 100 0 4.966 0.948 2.477 5.011 7.487
R2_demand 100 0 0.427 0.022 0.380 0.429 0.481
R2_supply 100 0 0.970 0.001 0.967 0.970 0.973

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.040 1.033 8.325 9.983 14.154
alpha1_hat 100 0 0.732 3.527 −7.984 0.989 13.497
alpha2_hat 100 0 1.034 0.395 −0.040 0.982 2.165
alpha3_hat 100 0 1.018 0.228 0.547 1.015 2.066
gamma0_hat 100 0 1.858 1.269 −2.277 1.935 4.987
gamma1_hat 100 0 −0.112 9.245 −80.104 0.724 17.025
gamma2_hat 100 0 0.908 0.196 0.531 0.894 1.593
gamma3_hat 100 0 0.921 0.203 0.489 0.888 1.383
theta_hat 100 0 0.166 0.748 −1.590 0.091 6.015
theta_hat_t_stats 100 0 0.254 1.155 −3.406 0.313 3.346
R2_demand 100 0 0.442 0.271 −1.045 0.510 0.773
R2_supply 100 0 0.975 0.009 0.950 0.976 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.888 0.665 8.647 9.810 11.863
alpha1_hat 100 0 1.235 2.322 −4.598 1.410 5.977
alpha2_hat 100 0 0.950 0.277 0.235 0.969 1.804
alpha3_hat 100 0 0.979 0.162 0.634 0.967 1.363
gamma0_hat 100 0 1.513 0.910 −0.805 1.551 3.438
gamma1_hat 100 0 0.714 3.539 −16.174 1.195 9.154
gamma2_hat 100 0 0.935 0.132 0.616 0.941 1.374
gamma3_hat 100 0 0.956 0.149 0.599 0.920 1.345
theta_hat 100 0 0.172 0.392 −0.710 0.124 2.259
theta_hat_t_stats 100 0 0.454 1.122 −1.879 0.449 3.147
R2_demand 100 0 0.474 0.143 −0.081 0.508 0.740
R2_supply 100 0 0.971 0.007 0.946 0.973 0.985
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.989 0.402 9.067 9.978 10.821
alpha1_hat 100 0 0.892 1.662 −2.511 0.818 5.197
alpha2_hat 100 0 1.010 0.174 0.651 1.021 1.376
alpha3_hat 100 0 1.008 0.093 0.737 1.010 1.240
gamma0_hat 100 0 1.188 0.680 −1.066 1.294 2.492
gamma1_hat 100 0 1.157 1.641 −2.695 1.002 5.961
gamma2_hat 100 0 0.986 0.104 0.762 0.976 1.306
gamma3_hat 100 0 0.980 0.097 0.779 0.973 1.294
theta_hat 100 0 0.156 0.173 −0.180 0.146 0.583
theta_hat_t_stats 100 0 0.810 0.918 −1.189 0.846 2.683
R2_demand 100 0 0.475 0.105 0.002 0.487 0.678
R2_supply 100 0 0.969 0.005 0.945 0.970 0.980
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.988 0.171 9.578 9.975 10.431
alpha1_hat 100 0 0.821 0.715 −0.816 0.789 2.254
alpha2_hat 100 0 1.015 0.078 0.850 1.014 1.272
alpha3_hat 100 0 0.997 0.042 0.893 1.000 1.083
gamma0_hat 100 0 1.016 0.314 0.234 0.993 1.766
gamma1_hat 100 0 1.040 0.712 −0.632 1.083 2.881
gamma2_hat 100 0 0.998 0.045 0.874 1.000 1.101
gamma3_hat 100 0 0.997 0.044 0.891 1.001 1.109
theta_hat 100 0 0.196 0.072 0.034 0.201 0.367
theta_hat_t_stats 100 0 2.445 0.848 0.469 2.439 4.215
R2_demand 100 0 0.473 0.043 0.342 0.479 0.554
R2_supply 100 0 0.968 0.003 0.961 0.968 0.974
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.996 0.145 9.645 9.980 10.376
alpha1_hat 100 0 1.038 0.491 0.048 1.017 2.282
alpha2_hat 100 0 0.995 0.058 0.795 0.986 1.122
alpha3_hat 100 0 1.000 0.032 0.908 1.001 1.091
gamma0_hat 100 0 0.988 0.214 0.340 1.013 1.465
gamma1_hat 100 0 0.996 0.594 −0.310 0.984 2.452
gamma2_hat 100 0 1.002 0.033 0.928 0.997 1.077
gamma3_hat 100 0 0.998 0.031 0.906 1.002 1.107
theta_hat 100 0 0.203 0.060 0.053 0.204 0.342
theta_hat_t_stats 100 0 3.497 0.936 1.053 3.489 5.619
R2_demand 100 0 0.479 0.035 0.380 0.483 0.534
R2_supply 100 0 0.967 0.002 0.963 0.967 0.971
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.080 9.850 9.989 10.241
alpha1_hat 100 0 0.987 0.308 0.308 0.955 2.050
alpha2_hat 100 0 0.999 0.032 0.921 1.004 1.083
alpha3_hat 100 0 0.999 0.020 0.950 0.998 1.046
gamma0_hat 100 0 1.024 0.132 0.669 1.022 1.312
gamma1_hat 100 0 1.036 0.344 0.092 1.070 1.812
gamma2_hat 100 0 0.999 0.020 0.961 0.996 1.047
gamma3_hat 100 0 0.996 0.020 0.938 0.997 1.047
theta_hat 100 0 0.193 0.035 0.102 0.193 0.282
theta_hat_t_stats 100 0 5.375 0.850 2.975 5.387 7.432
R2_demand 100 0 0.472 0.020 0.413 0.475 0.513
R2_supply 100 0 0.968 0.001 0.964 0.968 0.970

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.002 1.058 7.983 9.898 14.800
alpha1_hat 100 0 1.377 15.017 −48.230 0.894 38.341
alpha2_hat 100 0 4.940 1.918 0.544 4.849 15.191
alpha3_hat 100 0 1.006 0.242 0.572 0.991 2.311
gamma0_hat 100 0 1.668 1.318 −1.916 2.004 4.132
gamma1_hat 100 0 −3.592 24.851 −153.302 2.158 27.679
gamma2_hat 100 0 0.932 0.216 0.476 0.905 1.481
gamma3_hat 100 0 0.935 0.184 0.597 0.916 1.433
theta_hat 100 0 0.209 0.599 −0.779 0.092 3.676
theta_hat_t_stats 100 0 0.322 0.970 −3.092 0.301 2.834
R2_demand 100 0 0.506 0.292 −1.126 0.570 0.816
R2_supply 100 0 0.971 0.008 0.947 0.972 0.986
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.019 0.671 8.679 10.062 12.231
alpha1_hat 100 0 2.664 10.353 −18.538 1.161 33.037
alpha2_hat 100 0 4.837 1.178 1.936 4.857 7.383
alpha3_hat 100 0 1.002 0.139 0.698 0.996 1.357
gamma0_hat 100 0 1.537 0.867 −1.023 1.588 4.000
gamma1_hat 100 0 0.521 10.723 −30.732 1.315 35.311
gamma2_hat 100 0 0.934 0.145 0.534 0.925 1.421
gamma3_hat 100 0 0.936 0.148 0.528 0.942 1.247
theta_hat 100 0 0.147 0.253 −0.332 0.120 0.935
theta_hat_t_stats 100 0 0.529 0.912 −1.580 0.631 2.736
R2_demand 100 0 0.507 0.151 −0.049 0.532 0.741
R2_supply 100 0 0.969 0.007 0.948 0.970 0.982
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.955 0.387 9.235 9.913 11.116
alpha1_hat 100 0 0.953 6.612 −20.606 1.011 18.557
alpha2_hat 100 0 4.949 0.771 3.250 4.930 7.343
alpha3_hat 100 0 0.980 0.087 0.739 0.973 1.348
gamma0_hat 100 0 1.188 0.729 −0.856 1.261 2.496
gamma1_hat 100 0 0.058 6.724 −23.482 0.817 15.995
gamma2_hat 100 0 0.981 0.107 0.793 0.966 1.253
gamma3_hat 100 0 0.975 0.102 0.728 0.966 1.238
theta_hat 100 0 0.198 0.163 −0.159 0.188 0.745
theta_hat_t_stats 100 0 1.129 0.863 −1.178 1.131 2.988
R2_demand 100 0 0.521 0.102 0.148 0.532 0.682
R2_supply 100 0 0.966 0.007 0.948 0.967 0.977
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.982 0.193 9.560 9.971 10.435
alpha1_hat 100 0 1.092 3.330 −6.629 0.974 7.706
alpha2_hat 100 0 4.972 0.378 4.148 5.007 5.920
alpha3_hat 100 0 0.992 0.049 0.891 0.986 1.130
gamma0_hat 100 0 1.001 0.324 0.047 1.002 1.681
gamma1_hat 100 0 0.906 3.489 −7.750 0.661 10.978
gamma2_hat 100 0 1.000 0.046 0.892 1.003 1.121
gamma3_hat 100 0 1.007 0.046 0.895 1.004 1.110
theta_hat 100 0 0.203 0.073 0.021 0.211 0.369
theta_hat_t_stats 100 0 2.746 0.908 0.317 2.893 4.875
R2_demand 100 0 0.521 0.039 0.432 0.517 0.609
R2_supply 100 0 0.964 0.003 0.957 0.965 0.970
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.999 0.134 9.692 10.002 10.278
alpha1_hat 100 0 1.050 2.303 −6.558 1.026 6.791
alpha2_hat 100 0 4.995 0.248 4.188 4.964 5.820
alpha3_hat 100 0 1.002 0.030 0.929 1.001 1.072
gamma0_hat 100 0 1.006 0.214 0.455 1.007 1.479
gamma1_hat 100 0 0.794 2.162 −4.579 0.889 5.829
gamma2_hat 100 0 1.000 0.033 0.912 0.999 1.100
gamma3_hat 100 0 0.993 0.031 0.915 0.998 1.070
theta_hat 100 0 0.203 0.050 0.081 0.207 0.317
theta_hat_t_stats 100 0 3.932 0.883 1.737 4.060 5.907
R2_demand 100 0 0.519 0.029 0.437 0.521 0.587
R2_supply 100 0 0.964 0.002 0.959 0.964 0.970
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.084 9.818 10.003 10.194
alpha1_hat 100 0 0.896 1.299 −2.603 1.018 4.193
alpha2_hat 100 0 5.009 0.169 4.558 5.006 5.406
alpha3_hat 100 0 1.001 0.019 0.951 1.001 1.055
gamma0_hat 100 0 1.011 0.123 0.682 1.014 1.340
gamma1_hat 100 0 1.092 1.387 −2.121 1.167 4.265
gamma2_hat 100 0 0.998 0.020 0.954 0.998 1.051
gamma3_hat 100 0 0.997 0.018 0.956 0.997 1.037
theta_hat 100 0 0.197 0.034 0.110 0.196 0.306
theta_hat_t_stats 100 0 6.081 0.942 3.443 6.027 8.785
R2_demand 100 0 0.519 0.017 0.475 0.520 0.551
R2_supply 100 0 0.964 0.001 0.962 0.964 0.967

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.907 0.938 7.866 9.809 12.850
alpha1_hat 100 0 1.370 51.562 −120.993 −5.489 170.972
alpha2_hat 100 0 19.581 6.126 3.832 19.461 40.350
alpha3_hat 100 0 0.994 0.232 0.524 0.955 1.846
gamma0_hat 100 0 1.683 1.390 −4.084 1.832 4.899
gamma1_hat 100 0 3.355 66.792 −241.185 8.888 127.934
gamma2_hat 100 0 0.920 0.211 0.511 0.899 1.635
gamma3_hat 100 0 0.928 0.208 0.559 0.897 1.548
theta_hat 100 0 0.100 0.357 −0.602 0.056 1.401
theta_hat_t_stats 100 0 0.155 1.003 −2.517 0.197 2.386
R2_demand 100 0 0.504 0.230 −0.385 0.553 0.813
R2_supply 100 0 0.970 0.009 0.946 0.972 0.987
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.964 0.590 8.849 9.864 12.076
alpha1_hat 100 0 −6.842 38.993 −106.509 −8.018 94.391
alpha2_hat 100 0 20.716 4.666 11.087 20.285 34.440
alpha3_hat 100 0 0.994 0.129 0.703 0.993 1.496
gamma0_hat 100 0 1.493 0.992 −2.796 1.660 3.206
gamma1_hat 100 0 2.038 40.869 −135.476 6.841 171.344
gamma2_hat 100 0 0.949 0.150 0.641 0.925 1.503
gamma3_hat 100 0 0.926 0.140 0.638 0.925 1.406
theta_hat 100 0 0.130 0.233 −0.622 0.108 0.939
theta_hat_t_stats 100 0 0.529 0.930 −2.184 0.503 2.702
R2_demand 100 0 0.526 0.168 −0.418 0.558 0.786
R2_supply 100 0 0.968 0.008 0.946 0.970 0.981
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.041 0.450 9.145 10.047 11.300
alpha1_hat 100 0 3.898 32.444 −76.175 2.590 80.157
alpha2_hat 100 0 19.927 3.680 12.924 19.748 32.442
alpha3_hat 100 0 1.025 0.102 0.792 1.028 1.270
gamma0_hat 100 0 1.285 0.590 −0.167 1.296 2.716
gamma1_hat 100 0 −4.167 25.855 −69.564 −2.896 51.897
gamma2_hat 100 0 0.963 0.090 0.697 0.953 1.232
gamma3_hat 100 0 0.976 0.097 0.711 0.973 1.201
theta_hat 100 0 0.191 0.146 −0.222 0.189 0.615
theta_hat_t_stats 100 0 1.133 0.781 −1.341 1.176 2.605
R2_demand 100 0 0.517 0.094 0.251 0.529 0.722
R2_supply 100 0 0.966 0.006 0.943 0.966 0.976
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.986 0.172 9.642 9.959 10.463
alpha1_hat 100 0 1.846 13.554 −32.313 1.997 35.088
alpha2_hat 100 0 19.849 1.481 16.789 19.801 23.422
alpha3_hat 100 0 0.995 0.044 0.904 0.992 1.124
gamma0_hat 100 0 1.093 0.264 0.274 1.111 1.657
gamma1_hat 100 0 −0.558 12.284 −27.036 −2.192 33.117
gamma2_hat 100 0 0.991 0.039 0.910 0.985 1.098
gamma3_hat 100 0 0.991 0.045 0.898 0.991 1.129
theta_hat 100 0 0.197 0.071 0.048 0.204 0.367
theta_hat_t_stats 100 0 2.727 0.943 0.738 2.928 4.729
R2_demand 100 0 0.531 0.041 0.385 0.536 0.609
R2_supply 100 0 0.964 0.002 0.956 0.965 0.970
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.989 0.154 9.663 9.990 10.370
alpha1_hat 100 0 2.484 8.549 −26.399 3.687 21.425
alpha2_hat 100 0 19.802 1.119 17.548 19.748 23.647
alpha3_hat 100 0 0.999 0.034 0.918 1.000 1.075
gamma0_hat 100 0 1.019 0.191 0.570 1.010 1.504
gamma1_hat 100 0 0.937 9.375 −23.896 0.459 20.997
gamma2_hat 100 0 0.999 0.029 0.915 0.999 1.085
gamma3_hat 100 0 0.995 0.031 0.926 0.996 1.059
theta_hat 100 0 0.198 0.052 0.078 0.196 0.326
theta_hat_t_stats 100 0 3.868 0.898 1.648 3.929 5.755
R2_demand 100 0 0.530 0.029 0.470 0.531 0.594
R2_supply 100 0 0.964 0.002 0.958 0.964 0.968
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.092 9.755 10.000 10.283
alpha1_hat 100 0 0.720 5.440 −13.112 0.422 12.155
alpha2_hat 100 0 20.050 0.645 18.084 20.056 21.485
alpha3_hat 100 0 1.003 0.022 0.949 1.003 1.072
gamma0_hat 100 0 1.000 0.132 0.641 1.005 1.282
gamma1_hat 100 0 1.487 5.416 −12.258 1.872 12.291
gamma2_hat 100 0 0.999 0.019 0.957 0.999 1.052
gamma3_hat 100 0 1.000 0.020 0.959 1.000 1.060
theta_hat 100 0 0.198 0.032 0.121 0.194 0.291
theta_hat_t_stats 100 0 6.223 0.916 3.846 6.121 8.644
R2_demand 100 0 0.530 0.021 0.467 0.532 0.578
R2_supply 100 0 0.963 0.001 0.960 0.963 0.967

\(\theta = 0.33\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.101 0.903 7.916 10.022 12.702
alpha1_hat 100 0 1.053 1.063 −1.100 0.896 4.684
alpha2_hat 100 0 0.102 0.093 −0.171 0.098 0.323
alpha3_hat 100 0 1.003 0.219 0.569 0.991 1.623
gamma0_hat 100 0 2.101 1.182 −2.239 2.345 4.339
gamma1_hat 100 0 −0.057 8.699 −78.621 1.111 13.057
gamma2_hat 100 0 0.871 0.178 0.379 0.865 1.505
gamma3_hat 100 0 0.865 0.162 0.479 0.848 1.358
theta_hat 100 0 0.548 3.640 −5.918 0.066 30.956
theta_hat_t_stats 100 0 0.039 0.976 −2.689 0.096 2.392
R2_demand 100 0 0.169 0.305 −0.935 0.247 0.609
R2_supply 100 0 0.983 0.006 0.961 0.984 0.993
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.942 0.645 8.511 9.888 11.803
alpha1_hat 100 0 1.035 0.659 −1.098 1.009 2.949
alpha2_hat 100 0 0.094 0.060 −0.039 0.092 0.305
alpha3_hat 100 0 1.003 0.158 0.666 0.989 1.558
gamma0_hat 100 0 1.488 1.034 −2.473 1.614 3.378
gamma1_hat 100 0 1.266 5.449 −18.722 0.978 34.323
gamma2_hat 100 0 0.944 0.156 0.633 0.911 1.413
gamma3_hat 100 0 0.950 0.155 0.598 0.941 1.518
theta_hat 100 0 0.089 2.775 −17.028 0.265 10.366
theta_hat_t_stats 100 0 0.399 0.970 −2.346 0.332 2.162
R2_demand 100 0 0.245 0.153 −0.357 0.283 0.542
R2_supply 100 0 0.981 0.005 0.960 0.981 0.989
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.021 0.418 9.172 9.970 11.266
alpha1_hat 100 0 1.039 0.462 −0.247 0.990 2.050
alpha2_hat 100 0 0.097 0.042 −0.003 0.096 0.235
alpha3_hat 100 0 1.008 0.096 0.801 1.004 1.253
gamma0_hat 100 0 1.130 0.601 −0.869 1.166 2.449
gamma1_hat 100 0 0.579 1.482 −10.219 0.838 3.872
gamma2_hat 100 0 0.982 0.088 0.793 0.970 1.227
gamma3_hat 100 0 0.995 0.093 0.772 0.996 1.195
theta_hat 100 0 0.518 0.734 −0.915 0.381 5.753
theta_hat_t_stats 100 0 0.862 0.760 −0.845 0.878 2.298
R2_demand 100 0 0.230 0.134 −0.151 0.238 0.537
R2_supply 100 0 0.979 0.004 0.968 0.980 0.986
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.987 0.203 9.377 9.972 10.663
alpha1_hat 100 0 1.005 0.200 0.332 1.003 1.444
alpha2_hat 100 0 0.099 0.020 0.053 0.101 0.155
alpha3_hat 100 0 1.004 0.048 0.843 1.004 1.118
gamma0_hat 100 0 1.061 0.302 0.261 1.093 1.723
gamma1_hat 100 0 0.967 0.373 −0.108 0.958 1.843
gamma2_hat 100 0 0.989 0.049 0.863 0.990 1.119
gamma3_hat 100 0 1.003 0.038 0.925 1.000 1.099
theta_hat 100 0 0.341 0.195 −0.169 0.323 1.040
theta_hat_t_stats 100 0 1.725 0.932 −0.947 1.775 3.839
R2_demand 100 0 0.241 0.058 0.101 0.249 0.371
R2_supply 100 0 0.979 0.002 0.975 0.979 0.982
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.010 0.136 9.632 10.018 10.284
alpha1_hat 100 0 1.010 0.130 0.719 1.001 1.343
alpha2_hat 100 0 0.099 0.014 0.067 0.098 0.142
alpha3_hat 100 0 0.998 0.033 0.931 0.999 1.087
gamma0_hat 100 0 1.036 0.191 0.456 1.067 1.415
gamma1_hat 100 0 0.947 0.294 0.128 0.942 1.845
gamma2_hat 100 0 0.995 0.031 0.923 0.994 1.101
gamma3_hat 100 0 0.995 0.027 0.940 0.994 1.074
theta_hat 100 0 0.348 0.148 −0.136 0.349 0.762
theta_hat_t_stats 100 0 2.548 1.056 −1.061 2.560 4.884
R2_demand 100 0 0.230 0.042 0.126 0.223 0.333
R2_supply 100 0 0.979 0.001 0.975 0.979 0.981
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.996 0.094 9.756 10.009 10.186
alpha1_hat 100 0 0.984 0.093 0.700 0.981 1.208
alpha2_hat 100 0 0.101 0.009 0.079 0.101 0.121
alpha3_hat 100 0 1.001 0.021 0.948 1.002 1.060
gamma0_hat 100 0 1.003 0.132 0.608 1.002 1.282
gamma1_hat 100 0 1.053 0.172 0.592 1.059 1.458
gamma2_hat 100 0 0.999 0.020 0.950 1.000 1.050
gamma3_hat 100 0 0.999 0.019 0.958 0.998 1.044
theta_hat 100 0 0.304 0.088 0.057 0.295 0.533
theta_hat_t_stats 100 0 3.605 0.970 0.786 3.611 5.999
R2_demand 100 0 0.240 0.027 0.161 0.242 0.292
R2_supply 100 0 0.978 0.001 0.976 0.978 0.980

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.788 0.869 7.639 9.751 13.200
alpha1_hat 100 0 0.846 2.077 −3.651 0.938 6.905
alpha2_hat 100 0 0.484 0.233 −0.058 0.490 1.293
alpha3_hat 100 0 0.971 0.219 0.492 0.951 1.737
gamma0_hat 100 0 1.940 1.060 −0.552 1.890 5.341
gamma1_hat 100 0 −0.471 23.062 −142.335 1.086 106.108
gamma2_hat 100 0 0.904 0.167 0.484 0.909 1.337
gamma3_hat 100 0 0.893 0.161 0.500 0.882 1.366
theta_hat 100 0 0.472 4.310 −15.572 0.146 27.086
theta_hat_t_stats 100 0 0.373 1.021 −3.885 0.354 2.674
R2_demand 100 0 0.366 0.273 −1.213 0.436 0.762
R2_supply 100 0 0.980 0.006 0.962 0.981 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.124 0.601 8.625 10.097 11.521
alpha1_hat 100 0 0.972 1.435 −3.407 0.881 5.203
alpha2_hat 100 0 0.523 0.162 0.116 0.529 0.973
alpha3_hat 100 0 1.044 0.148 0.716 1.051 1.519
gamma0_hat 100 0 1.615 0.782 0.077 1.713 3.406
gamma1_hat 100 0 1.036 1.866 −4.931 0.967 7.421
gamma2_hat 100 0 0.939 0.119 0.665 0.942 1.230
gamma3_hat 100 0 0.932 0.132 0.525 0.914 1.323
theta_hat 100 0 0.217 0.317 −0.681 0.195 1.296
theta_hat_t_stats 100 0 0.627 0.797 −1.411 0.785 2.106
R2_demand 100 0 0.314 0.194 −0.334 0.351 0.618
R2_supply 100 0 0.978 0.005 0.962 0.978 0.988
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.969 0.451 8.923 9.933 11.168
alpha1_hat 100 0 0.994 1.062 −2.433 0.846 4.063
alpha2_hat 100 0 0.496 0.109 0.221 0.488 0.786
alpha3_hat 100 0 0.999 0.116 0.726 1.009 1.273
gamma0_hat 100 0 1.412 0.554 −0.066 1.395 2.680
gamma1_hat 100 0 0.886 1.509 −2.706 0.999 4.033
gamma2_hat 100 0 0.950 0.082 0.738 0.945 1.196
gamma3_hat 100 0 0.968 0.101 0.715 0.958 1.269
theta_hat 100 0 0.288 0.259 −0.218 0.250 0.960
theta_hat_t_stats 100 0 1.221 1.021 −0.736 1.172 3.682
R2_demand 100 0 0.357 0.115 0.000 0.379 0.556
R2_supply 100 0 0.976 0.004 0.965 0.976 0.984
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.989 0.216 9.527 9.985 10.570
alpha1_hat 100 0 0.921 0.428 −0.146 0.935 2.314
alpha2_hat 100 0 0.506 0.054 0.332 0.504 0.620
alpha3_hat 100 0 1.000 0.042 0.899 1.000 1.119
gamma0_hat 100 0 1.070 0.286 −0.021 1.105 1.625
gamma1_hat 100 0 1.045 0.536 −0.108 0.981 2.247
gamma2_hat 100 0 0.994 0.042 0.892 0.996 1.115
gamma3_hat 100 0 0.994 0.042 0.886 0.989 1.093
theta_hat 100 0 0.311 0.101 0.089 0.312 0.535
theta_hat_t_stats 100 0 3.146 0.911 0.874 3.321 5.127
R2_demand 100 0 0.355 0.058 0.160 0.364 0.475
R2_supply 100 0 0.974 0.002 0.968 0.974 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.983 0.126 9.699 9.986 10.273
alpha1_hat 100 0 0.951 0.318 0.161 0.927 1.761
alpha2_hat 100 0 0.503 0.034 0.427 0.500 0.587
alpha3_hat 100 0 0.996 0.031 0.916 0.995 1.082
gamma0_hat 100 0 1.006 0.213 0.469 1.036 1.513
gamma1_hat 100 0 1.010 0.382 0.047 1.015 1.956
gamma2_hat 100 0 1.003 0.030 0.931 1.002 1.087
gamma3_hat 100 0 0.997 0.031 0.920 1.001 1.070
theta_hat 100 0 0.328 0.078 0.153 0.323 0.524
theta_hat_t_stats 100 0 4.625 0.949 2.228 4.641 6.600
R2_demand 100 0 0.366 0.031 0.286 0.368 0.439
R2_supply 100 0 0.973 0.001 0.970 0.973 0.977
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.085 9.817 9.998 10.200
alpha1_hat 100 0 1.034 0.202 0.571 1.027 1.492
alpha2_hat 100 0 0.497 0.022 0.452 0.499 0.546
alpha3_hat 100 0 1.000 0.021 0.946 1.000 1.058
gamma0_hat 100 0 1.022 0.123 0.705 1.025 1.322
gamma1_hat 100 0 0.929 0.257 0.225 0.961 1.735
gamma2_hat 100 0 0.998 0.017 0.945 0.999 1.039
gamma3_hat 100 0 0.996 0.019 0.953 0.997 1.043
theta_hat 100 0 0.338 0.048 0.167 0.338 0.473
theta_hat_t_stats 100 0 7.552 0.945 3.742 7.527 9.552
R2_demand 100 0 0.359 0.021 0.309 0.361 0.424
R2_supply 100 0 0.973 0.001 0.971 0.973 0.976

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.881 0.911 7.719 9.780 12.297
alpha1_hat 100 0 0.820 3.832 −10.976 1.218 10.451
alpha2_hat 100 0 0.987 0.444 0.155 0.988 2.274
alpha3_hat 100 0 0.974 0.233 0.323 0.985 1.659
gamma0_hat 100 0 1.911 1.300 −3.608 2.020 4.784
gamma1_hat 100 0 0.090 7.240 −30.114 0.326 47.269
gamma2_hat 100 0 0.908 0.203 0.463 0.914 1.607
gamma3_hat 100 0 0.917 0.195 0.494 0.922 1.646
theta_hat 100 0 0.300 0.720 −3.238 0.197 3.616
theta_hat_t_stats 100 0 0.506 0.984 −2.270 0.543 2.596
R2_demand 100 0 0.433 0.229 −0.605 0.487 0.756
R2_supply 100 0 0.978 0.007 0.949 0.979 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.911 0.619 8.764 9.845 11.944
alpha1_hat 100 0 0.851 2.377 −4.596 0.806 6.231
alpha2_hat 100 0 0.991 0.288 0.353 0.978 1.800
alpha3_hat 100 0 0.999 0.152 0.641 0.992 1.480
gamma0_hat 100 0 1.488 0.872 −1.487 1.554 3.563
gamma1_hat 100 0 1.040 3.080 −7.492 1.572 8.048
gamma2_hat 100 0 0.947 0.132 0.676 0.935 1.367
gamma3_hat 100 0 0.935 0.140 0.661 0.934 1.404
theta_hat 100 0 0.259 0.320 −0.370 0.226 1.049
theta_hat_t_stats 100 0 0.835 0.964 −1.636 0.806 3.122
R2_demand 100 0 0.402 0.166 −0.199 0.436 0.642
R2_supply 100 0 0.975 0.005 0.961 0.975 0.985
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.991 0.399 9.100 9.926 10.992
alpha1_hat 100 0 0.978 1.600 −2.596 1.086 4.918
alpha2_hat 100 0 1.004 0.184 0.606 0.988 1.468
alpha3_hat 100 0 0.998 0.087 0.807 0.986 1.228
gamma0_hat 100 0 1.280 0.648 −0.417 1.291 3.683
gamma1_hat 100 0 1.116 2.008 −7.273 1.252 6.573
gamma2_hat 100 0 0.962 0.099 0.622 0.956 1.209
gamma3_hat 100 0 0.970 0.095 0.701 0.966 1.229
theta_hat 100 0 0.280 0.217 −0.146 0.289 1.150
theta_hat_t_stats 100 0 1.337 0.925 −0.873 1.428 3.419
R2_demand 100 0 0.377 0.113 0.060 0.406 0.605
R2_supply 100 0 0.973 0.004 0.956 0.974 0.981
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.009 0.199 9.552 10.022 10.515
alpha1_hat 100 0 1.064 0.807 −1.336 1.050 3.026
alpha2_hat 100 0 0.997 0.092 0.774 0.993 1.273
alpha3_hat 100 0 1.006 0.052 0.901 1.006 1.140
gamma0_hat 100 0 1.051 0.270 0.312 1.040 1.585
gamma1_hat 100 0 0.966 0.860 −1.327 0.975 2.861
gamma2_hat 100 0 0.993 0.039 0.924 0.986 1.088
gamma3_hat 100 0 0.998 0.041 0.908 0.999 1.099
theta_hat 100 0 0.325 0.094 0.123 0.313 0.627
theta_hat_t_stats 100 0 3.614 0.900 1.572 3.584 5.929
R2_demand 100 0 0.392 0.048 0.264 0.393 0.500
R2_supply 100 0 0.972 0.002 0.965 0.972 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.983 0.130 9.671 9.990 10.252
alpha1_hat 100 0 1.070 0.629 −0.608 1.069 2.884
alpha2_hat 100 0 0.989 0.064 0.812 0.983 1.179
alpha3_hat 100 0 0.996 0.032 0.915 0.996 1.087
gamma0_hat 100 0 1.008 0.229 0.359 1.030 1.458
gamma1_hat 100 0 0.987 0.636 −0.511 1.017 2.559
gamma2_hat 100 0 0.999 0.034 0.935 0.996 1.089
gamma3_hat 100 0 0.999 0.037 0.931 0.993 1.102
theta_hat 100 0 0.332 0.064 0.190 0.323 0.479
theta_hat_t_stats 100 0 5.206 0.859 3.152 5.072 7.260
R2_demand 100 0 0.400 0.034 0.302 0.401 0.492
R2_supply 100 0 0.972 0.002 0.966 0.972 0.975
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.015 0.085 9.826 10.024 10.269
alpha1_hat 100 0 1.057 0.391 −0.222 1.003 2.088
alpha2_hat 100 0 0.997 0.043 0.904 0.998 1.133
alpha3_hat 100 0 1.003 0.020 0.961 1.002 1.064
gamma0_hat 100 0 1.017 0.132 0.663 1.020 1.394
gamma1_hat 100 0 0.983 0.396 0.141 0.973 1.872
gamma2_hat 100 0 0.998 0.019 0.948 0.999 1.045
gamma3_hat 100 0 0.999 0.023 0.944 1.000 1.057
theta_hat 100 0 0.329 0.040 0.227 0.330 0.434
theta_hat_t_stats 100 0 8.273 0.926 5.624 8.335 10.763
R2_demand 100 0 0.395 0.020 0.347 0.395 0.445
R2_supply 100 0 0.972 0.001 0.970 0.972 0.974

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.155 1.110 8.022 9.940 14.074
alpha1_hat 100 0 −0.743 17.362 −47.100 −1.777 52.841
alpha2_hat 100 0 5.352 1.980 0.878 5.258 10.056
alpha3_hat 100 0 1.028 0.251 0.492 0.996 2.140
gamma0_hat 100 0 2.017 1.255 −5.510 2.302 4.693
gamma1_hat 100 0 1.114 20.733 −95.941 2.949 59.162
gamma2_hat 100 0 0.865 0.199 0.450 0.854 1.963
gamma3_hat 100 0 0.911 0.234 0.528 0.885 2.352
theta_hat 100 0 0.222 0.481 −0.764 0.158 2.564
theta_hat_t_stats 100 0 0.397 0.903 −1.824 0.476 2.470
R2_demand 100 0 0.344 0.381 −1.444 0.452 0.757
R2_supply 100 0 0.976 0.008 0.921 0.977 0.991
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.034 0.656 8.761 9.962 12.255
alpha1_hat 100 0 1.650 12.396 −33.550 0.718 38.354
alpha2_hat 100 0 4.988 1.422 2.300 4.944 9.145
alpha3_hat 100 0 1.001 0.156 0.577 1.020 1.546
gamma0_hat 100 0 1.471 0.939 −1.018 1.598 3.391
gamma1_hat 100 0 −1.200 13.627 −38.054 −0.765 29.000
gamma2_hat 100 0 0.961 0.146 0.672 0.944 1.371
gamma3_hat 100 0 0.932 0.131 0.658 0.929 1.359
theta_hat 100 0 0.300 0.298 −0.258 0.258 1.403
theta_hat_t_stats 100 0 1.022 0.897 −1.367 1.095 2.478
R2_demand 100 0 0.424 0.158 −0.076 0.453 0.721
R2_supply 100 0 0.973 0.006 0.951 0.975 0.985
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.080 0.461 9.105 10.044 11.438
alpha1_hat 100 0 2.168 8.411 −18.957 2.191 21.653
alpha2_hat 100 0 4.968 0.909 2.360 4.973 7.329
alpha3_hat 100 0 1.003 0.094 0.807 0.992 1.269
gamma0_hat 100 0 1.250 0.599 −0.275 1.324 2.542
gamma1_hat 100 0 −1.451 9.317 −31.053 −0.677 20.041
gamma2_hat 100 0 0.967 0.096 0.739 0.954 1.175
gamma3_hat 100 0 0.971 0.099 0.753 0.963 1.242
theta_hat 100 0 0.341 0.224 −0.166 0.310 1.125
theta_hat_t_stats 100 0 1.772 0.928 −0.992 1.948 3.908
R2_demand 100 0 0.424 0.107 0.069 0.441 0.604
R2_supply 100 0 0.972 0.005 0.962 0.972 0.980
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.017 0.207 9.502 10.010 10.834
alpha1_hat 100 0 1.106 3.724 −11.821 0.780 11.378
alpha2_hat 100 0 5.009 0.401 4.036 5.032 6.438
alpha3_hat 100 0 0.998 0.050 0.890 1.000 1.130
gamma0_hat 100 0 1.016 0.328 0.261 1.029 1.662
gamma1_hat 100 0 0.826 4.321 −10.111 1.285 8.960
gamma2_hat 100 0 0.999 0.051 0.879 0.996 1.125
gamma3_hat 100 0 0.994 0.046 0.884 0.989 1.087
theta_hat 100 0 0.331 0.094 0.116 0.318 0.561
theta_hat_t_stats 100 0 4.061 0.998 1.572 3.992 6.288
R2_demand 100 0 0.435 0.049 0.297 0.443 0.561
R2_supply 100 0 0.969 0.003 0.960 0.969 0.974
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.146 9.661 10.001 10.377
alpha1_hat 100 0 0.880 2.928 −9.563 0.832 8.123
alpha2_hat 100 0 5.011 0.310 4.243 5.014 6.091
alpha3_hat 100 0 0.998 0.030 0.908 0.999 1.086
gamma0_hat 100 0 1.041 0.230 0.513 1.059 1.549
gamma1_hat 100 0 0.828 2.831 −6.201 0.521 7.237
gamma2_hat 100 0 0.996 0.034 0.900 1.001 1.080
gamma3_hat 100 0 0.996 0.032 0.925 0.993 1.093
theta_hat 100 0 0.327 0.069 0.170 0.318 0.482
theta_hat_t_stats 100 0 5.752 0.969 3.216 5.735 7.801
R2_demand 100 0 0.437 0.032 0.337 0.438 0.501
R2_supply 100 0 0.970 0.002 0.965 0.970 0.975
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.088 9.713 9.995 10.212
alpha1_hat 100 0 0.987 1.683 −3.239 0.900 5.071
alpha2_hat 100 0 4.998 0.194 4.540 4.998 5.521
alpha3_hat 100 0 1.001 0.020 0.962 1.003 1.053
gamma0_hat 100 0 1.028 0.153 0.682 1.024 1.527
gamma1_hat 100 0 1.088 1.592 −2.786 1.001 5.915
gamma2_hat 100 0 0.998 0.021 0.933 1.000 1.046
gamma3_hat 100 0 0.996 0.022 0.948 0.996 1.053
theta_hat 100 0 0.324 0.035 0.233 0.326 0.421
theta_hat_t_stats 100 0 9.026 0.783 6.759 9.111 10.704
R2_demand 100 0 0.439 0.023 0.391 0.439 0.495
R2_supply 100 0 0.970 0.001 0.966 0.970 0.972

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.012 1.402 7.889 9.766 20.673
alpha1_hat 100 0 7.425 85.038 −130.833 7.690 601.521
alpha2_hat 100 0 19.400 7.157 3.583 18.262 41.696
alpha3_hat 100 0 1.006 0.267 0.477 0.984 2.735
gamma0_hat 100 0 1.946 1.099 −0.612 1.987 5.051
gamma1_hat 100 0 −24.083 84.554 −333.040 −2.962 124.982
gamma2_hat 100 0 0.899 0.178 0.420 0.901 1.345
gamma3_hat 100 0 0.922 0.185 0.432 0.909 1.476
theta_hat 100 0 0.324 0.513 −0.388 0.204 2.383
theta_hat_t_stats 100 0 0.621 0.897 −1.148 0.597 2.443
R2_demand 100 0 0.371 0.875 −7.955 0.533 0.791
R2_supply 100 0 0.976 0.006 0.958 0.976 0.988
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.079 0.710 8.357 9.956 13.374
alpha1_hat 100 0 −5.827 50.208 −119.702 1.818 123.228
alpha2_hat 100 0 20.975 6.353 4.808 19.588 36.780
alpha3_hat 100 0 1.019 0.150 0.700 0.997 1.424
gamma0_hat 100 0 1.407 0.899 −1.648 1.417 3.217
gamma1_hat 100 0 −2.339 66.081 −279.507 4.658 167.711
gamma2_hat 100 0 0.971 0.139 0.609 0.946 1.352
gamma3_hat 100 0 0.939 0.141 0.631 0.925 1.317
theta_hat 100 0 0.291 0.403 −0.522 0.212 2.469
theta_hat_t_stats 100 0 0.999 0.926 −1.255 0.983 4.162
R2_demand 100 0 0.415 0.203 −0.875 0.450 0.720
R2_supply 100 0 0.973 0.006 0.949 0.974 0.982
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.922 0.407 9.079 9.935 10.884
alpha1_hat 100 0 −1.508 28.924 −68.330 −3.907 68.355
alpha2_hat 100 0 19.881 3.396 11.564 19.740 28.292
alpha3_hat 100 0 0.982 0.101 0.768 0.984 1.289
gamma0_hat 100 0 1.256 0.637 −0.804 1.330 2.795
gamma1_hat 100 0 3.535 36.126 −90.736 2.706 103.066
gamma2_hat 100 0 0.968 0.097 0.707 0.953 1.293
gamma3_hat 100 0 0.974 0.106 0.651 0.983 1.249
theta_hat 100 0 0.291 0.202 −0.219 0.284 0.997
theta_hat_t_stats 100 0 1.550 0.943 −1.108 1.669 3.494
R2_demand 100 0 0.446 0.109 0.103 0.464 0.666
R2_supply 100 0 0.971 0.005 0.959 0.971 0.979
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.014 0.213 9.553 10.005 10.726
alpha1_hat 100 0 −0.823 13.567 −28.407 −2.822 40.858
alpha2_hat 100 0 20.238 1.701 16.009 20.199 24.180
alpha3_hat 100 0 1.003 0.048 0.920 1.002 1.161
gamma0_hat 100 0 1.048 0.279 0.243 1.084 1.593
gamma1_hat 100 0 0.711 14.816 −58.674 2.382 35.044
gamma2_hat 100 0 0.994 0.046 0.907 0.996 1.103
gamma3_hat 100 0 1.003 0.042 0.911 0.997 1.127
theta_hat 100 0 0.325 0.085 0.124 0.324 0.682
theta_hat_t_stats 100 0 4.106 0.892 2.049 4.137 6.990
R2_demand 100 0 0.450 0.053 0.302 0.458 0.542
R2_supply 100 0 0.969 0.002 0.963 0.969 0.973
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.988 0.137 9.717 9.973 10.406
alpha1_hat 100 0 1.148 10.368 −31.475 0.356 30.391
alpha2_hat 100 0 19.959 1.206 17.236 19.892 23.249
alpha3_hat 100 0 0.999 0.031 0.925 0.999 1.086
gamma0_hat 100 0 0.990 0.196 0.581 0.969 1.570
gamma1_hat 100 0 1.402 9.730 −25.260 1.292 23.032
gamma2_hat 100 0 1.001 0.028 0.905 0.998 1.066
gamma3_hat 100 0 0.998 0.033 0.919 0.998 1.063
theta_hat 100 0 0.330 0.057 0.210 0.329 0.492
theta_hat_t_stats 100 0 5.871 0.860 4.021 5.787 7.935
R2_demand 100 0 0.446 0.035 0.350 0.447 0.527
R2_supply 100 0 0.969 0.001 0.965 0.969 0.973
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.098 9.752 10.008 10.213
alpha1_hat 100 0 1.032 6.687 −12.985 1.126 13.148
alpha2_hat 100 0 20.003 0.827 17.757 19.938 22.084
alpha3_hat 100 0 1.001 0.022 0.938 1.002 1.048
gamma0_hat 100 0 1.022 0.140 0.704 1.028 1.322
gamma1_hat 100 0 −0.175 7.039 −20.876 −0.177 18.972
gamma2_hat 100 0 0.998 0.021 0.959 0.999 1.050
gamma3_hat 100 0 0.998 0.020 0.954 0.999 1.044
theta_hat 100 0 0.333 0.043 0.196 0.330 0.480
theta_hat_t_stats 100 0 9.418 0.958 6.142 9.391 12.252
R2_demand 100 0 0.447 0.022 0.389 0.449 0.490
R2_supply 100 0 0.969 0.001 0.967 0.969 0.971

\(\theta = 0.5\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.068 1.045 7.915 9.982 13.936
alpha1_hat 100 0 1.002 1.373 −1.912 1.049 7.658
alpha2_hat 100 0 0.104 0.129 −0.276 0.093 0.585
alpha3_hat 100 0 0.987 0.208 0.523 0.976 1.566
gamma0_hat 100 0 1.946 1.187 −1.071 2.128 4.491
gamma1_hat 100 0 3.282 22.714 −15.653 1.134 222.728
gamma2_hat 100 0 0.886 0.193 0.453 0.862 1.363
gamma3_hat 100 0 0.910 0.164 0.443 0.888 1.302
theta_hat 100 0 −0.523 9.168 −87.433 0.233 10.914
theta_hat_t_stats 100 0 0.260 1.073 −3.003 0.388 2.669
R2_demand 100 0 0.125 0.446 −2.641 0.237 0.556
R2_supply 100 0 0.985 0.005 0.965 0.986 0.993
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.942 0.620 8.837 9.822 12.284
alpha1_hat 100 0 1.077 0.863 −0.684 1.102 5.283
alpha2_hat 100 0 0.089 0.076 −0.161 0.093 0.261
alpha3_hat 100 0 0.994 0.148 0.680 0.990 1.396
gamma0_hat 100 0 1.494 0.996 −0.860 1.568 3.462
gamma1_hat 100 0 0.552 4.245 −27.833 0.871 12.577
gamma2_hat 100 0 0.941 0.155 0.600 0.928 1.323
gamma3_hat 100 0 0.953 0.131 0.685 0.949 1.341
theta_hat 100 0 0.663 2.217 −5.753 0.393 15.036
theta_hat_t_stats 100 0 0.646 1.046 −1.990 0.636 3.268
R2_demand 100 0 0.200 0.209 −0.678 0.235 0.613
R2_supply 100 0 0.982 0.005 0.964 0.984 0.989
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.972 0.468 8.912 9.949 10.966
alpha1_hat 100 0 0.955 0.510 −0.215 0.908 2.705
alpha2_hat 100 0 0.102 0.050 −0.020 0.098 0.219
alpha3_hat 100 0 0.983 0.101 0.773 0.981 1.267
gamma0_hat 100 0 1.239 0.669 −0.576 1.320 2.801
gamma1_hat 100 0 1.019 2.517 −7.050 0.920 20.405
gamma2_hat 100 0 0.974 0.100 0.693 0.962 1.281
gamma3_hat 100 0 0.975 0.090 0.782 0.973 1.235
theta_hat 100 0 0.453 1.466 −10.534 0.484 6.337
theta_hat_t_stats 100 0 0.970 1.071 −3.039 1.078 3.586
R2_demand 100 0 0.188 0.151 −0.202 0.204 0.447
R2_supply 100 0 0.981 0.004 0.968 0.981 0.988
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.197 9.598 9.982 10.443
alpha1_hat 100 0 0.975 0.235 0.360 0.962 1.452
alpha2_hat 100 0 0.102 0.023 0.059 0.100 0.187
alpha3_hat 100 0 1.001 0.046 0.915 0.997 1.101
gamma0_hat 100 0 1.101 0.299 0.312 1.118 1.830
gamma1_hat 100 0 1.069 0.455 −0.221 1.078 2.113
gamma2_hat 100 0 0.990 0.044 0.892 0.992 1.120
gamma3_hat 100 0 0.986 0.049 0.873 0.984 1.139
theta_hat 100 0 0.444 0.244 −0.098 0.426 1.195
theta_hat_t_stats 100 0 2.055 0.964 −0.536 1.938 3.985
R2_demand 100 0 0.195 0.066 0.021 0.200 0.320
R2_supply 100 0 0.981 0.001 0.977 0.981 0.983
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.139 9.666 10.013 10.278
alpha1_hat 100 0 0.995 0.176 0.583 0.985 1.451
alpha2_hat 100 0 0.101 0.017 0.064 0.102 0.144
alpha3_hat 100 0 1.005 0.030 0.930 1.005 1.080
gamma0_hat 100 0 0.997 0.215 0.509 1.016 1.508
gamma1_hat 100 0 0.931 0.345 0.053 0.984 1.585
gamma2_hat 100 0 1.001 0.034 0.931 1.001 1.077
gamma3_hat 100 0 0.998 0.031 0.930 0.997 1.103
theta_hat 100 0 0.533 0.173 0.235 0.504 0.975
theta_hat_t_stats 100 0 3.535 0.873 1.638 3.489 5.790
R2_demand 100 0 0.202 0.041 0.083 0.205 0.277
R2_supply 100 0 0.980 0.001 0.978 0.980 0.982
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.005 0.084 9.802 10.012 10.204
alpha1_hat 100 0 1.022 0.106 0.801 1.032 1.255
alpha2_hat 100 0 0.098 0.010 0.077 0.098 0.123
alpha3_hat 100 0 1.001 0.021 0.945 0.999 1.046
gamma0_hat 100 0 1.014 0.138 0.676 1.022 1.306
gamma1_hat 100 0 0.967 0.224 0.552 0.960 1.472
gamma2_hat 100 0 0.998 0.020 0.959 0.999 1.056
gamma3_hat 100 0 0.996 0.019 0.951 0.995 1.050
theta_hat 100 0 0.512 0.113 0.274 0.523 0.698
theta_hat_t_stats 100 0 5.361 1.001 3.055 5.452 7.626
R2_demand 100 0 0.200 0.028 0.140 0.198 0.277
R2_supply 100 0 0.980 0.001 0.978 0.980 0.982

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.918 1.030 7.960 9.700 14.079
alpha1_hat 100 0 0.635 2.503 −4.577 0.383 7.811
alpha2_hat 100 0 0.522 0.295 −0.166 0.537 1.309
alpha3_hat 100 0 0.993 0.217 0.623 0.971 1.780
gamma0_hat 100 0 1.758 1.224 −1.300 1.985 3.777
gamma1_hat 100 0 −1.045 16.220 −126.985 1.032 31.150
gamma2_hat 100 0 0.906 0.191 0.527 0.879 1.405
gamma3_hat 100 0 0.893 0.186 0.463 0.862 1.493
theta_hat 100 0 0.791 3.112 −5.263 0.320 22.229
theta_hat_t_stats 100 0 0.657 1.014 −2.406 0.576 3.168
R2_demand 100 0 0.262 0.375 −1.820 0.348 0.696
R2_supply 100 0 0.981 0.006 0.959 0.981 0.993
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.042 0.710 8.658 9.920 12.798
alpha1_hat 100 0 1.142 1.940 −3.704 0.923 7.108
alpha2_hat 100 0 0.494 0.197 −0.054 0.486 1.028
alpha3_hat 100 0 1.019 0.154 0.644 1.032 1.481
gamma0_hat 100 0 1.596 0.891 −0.837 1.664 3.490
gamma1_hat 100 0 0.752 5.439 −18.226 1.268 37.865
gamma2_hat 100 0 0.938 0.137 0.564 0.935 1.332
gamma3_hat 100 0 0.932 0.129 0.623 0.935 1.502
theta_hat 100 0 0.446 0.927 −5.465 0.336 4.173
theta_hat_t_stats 100 0 1.012 0.958 −2.080 1.031 2.929
R2_demand 100 0 0.260 0.243 −1.117 0.336 0.554
R2_supply 100 0 0.980 0.005 0.960 0.981 0.988
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.452 9.236 9.951 11.002
alpha1_hat 100 0 0.901 1.288 −2.563 0.945 4.009
alpha2_hat 100 0 0.511 0.126 0.169 0.517 0.786
alpha3_hat 100 0 0.992 0.093 0.735 0.992 1.356
gamma0_hat 100 0 1.272 0.618 −0.296 1.216 2.596
gamma1_hat 100 0 0.925 1.779 −7.980 1.108 5.833
gamma2_hat 100 0 0.967 0.098 0.778 0.970 1.302
gamma3_hat 100 0 0.964 0.103 0.767 0.967 1.295
theta_hat 100 0 0.469 0.331 −0.176 0.442 1.985
theta_hat_t_stats 100 0 1.777 0.876 −0.464 1.836 3.539
R2_demand 100 0 0.270 0.137 −0.168 0.277 0.526
R2_supply 100 0 0.978 0.004 0.965 0.978 0.986
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.985 0.190 9.511 9.980 10.607
alpha1_hat 100 0 1.057 0.577 −0.562 1.106 2.106
alpha2_hat 100 0 0.492 0.063 0.370 0.482 0.652
alpha3_hat 100 0 0.997 0.045 0.884 0.996 1.102
gamma0_hat 100 0 1.034 0.340 −0.018 1.038 2.149
gamma1_hat 100 0 0.893 0.748 −0.711 0.872 2.260
gamma2_hat 100 0 0.995 0.051 0.799 0.991 1.138
gamma3_hat 100 0 0.997 0.047 0.867 0.992 1.140
theta_hat 100 0 0.515 0.122 0.257 0.521 0.866
theta_hat_t_stats 100 0 4.488 0.899 2.388 4.525 6.811
R2_demand 100 0 0.292 0.056 0.164 0.300 0.415
R2_supply 100 0 0.977 0.002 0.972 0.977 0.982
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.997 0.154 9.672 9.988 10.373
alpha1_hat 100 0 0.951 0.370 0.116 0.988 2.276
alpha2_hat 100 0 0.505 0.040 0.409 0.505 0.618
alpha3_hat 100 0 1.001 0.034 0.914 0.997 1.070
gamma0_hat 100 0 0.992 0.228 0.432 0.991 1.426
gamma1_hat 100 0 1.005 0.520 −0.213 0.939 2.031
gamma2_hat 100 0 1.002 0.035 0.933 1.006 1.082
gamma3_hat 100 0 1.001 0.036 0.907 1.001 1.089
theta_hat 100 0 0.500 0.092 0.309 0.509 0.713
theta_hat_t_stats 100 0 6.391 0.966 4.223 6.466 9.099
R2_demand 100 0 0.286 0.046 0.149 0.289 0.386
R2_supply 100 0 0.977 0.001 0.973 0.977 0.979
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.085 9.719 10.004 10.156
alpha1_hat 100 0 1.011 0.213 0.549 1.025 1.559
alpha2_hat 100 0 0.499 0.024 0.432 0.501 0.555
alpha3_hat 100 0 0.998 0.019 0.940 0.999 1.048
gamma0_hat 100 0 1.005 0.147 0.522 1.013 1.289
gamma1_hat 100 0 1.042 0.333 −0.115 1.089 1.851
gamma2_hat 100 0 1.000 0.022 0.956 1.000 1.053
gamma3_hat 100 0 1.000 0.021 0.938 0.996 1.068
theta_hat 100 0 0.491 0.063 0.346 0.487 0.680
theta_hat_t_stats 100 0 9.850 1.016 7.246 9.825 13.226
R2_demand 100 0 0.287 0.024 0.233 0.286 0.368
R2_supply 100 0 0.976 0.001 0.974 0.976 0.978

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.025 1.054 8.416 9.819 13.799
alpha1_hat 100 0 1.174 4.966 −12.688 0.825 25.156
alpha2_hat 100 0 0.989 0.588 −0.632 0.949 2.952
alpha3_hat 100 0 1.004 0.258 0.489 0.952 1.923
gamma0_hat 100 0 2.020 1.221 −1.393 2.018 4.746
gamma1_hat 100 0 −3.453 22.831 −182.931 1.151 18.264
gamma2_hat 100 0 0.901 0.189 0.498 0.902 1.446
gamma3_hat 100 0 0.882 0.184 0.476 0.903 1.459
theta_hat 100 0 0.810 2.454 −1.120 0.249 21.182
theta_hat_t_stats 100 0 0.670 0.954 −2.071 0.658 3.032
R2_demand 100 0 0.275 0.424 −2.122 0.397 0.744
R2_supply 100 0 0.981 0.006 0.961 0.983 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.014 0.683 8.583 9.864 12.234
alpha1_hat 100 0 0.899 2.964 −7.428 0.718 8.488
alpha2_hat 100 0 1.017 0.314 0.322 1.005 2.141
alpha3_hat 100 0 0.993 0.136 0.714 0.979 1.387
gamma0_hat 100 0 1.450 0.844 −1.056 1.371 3.667
gamma1_hat 100 0 0.700 4.041 −13.033 1.743 7.265
gamma2_hat 100 0 0.956 0.133 0.679 0.959 1.328
gamma3_hat 100 0 0.947 0.128 0.644 0.939 1.224
theta_hat 100 0 0.446 0.398 −0.250 0.356 2.145
theta_hat_t_stats 100 0 1.318 0.915 −1.169 1.256 3.463
R2_demand 100 0 0.304 0.191 −0.249 0.329 0.608
R2_supply 100 0 0.979 0.004 0.958 0.979 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.052 0.488 8.847 10.022 11.306
alpha1_hat 100 0 1.198 2.177 −3.931 1.367 6.972
alpha2_hat 100 0 0.993 0.241 0.512 0.976 1.761
alpha3_hat 100 0 1.016 0.106 0.780 1.002 1.317
gamma0_hat 100 0 1.143 0.680 −1.771 1.160 2.630
gamma1_hat 100 0 0.612 2.660 −5.428 0.975 5.410
gamma2_hat 100 0 0.979 0.098 0.762 0.979 1.401
gamma3_hat 100 0 0.984 0.096 0.800 0.972 1.286
theta_hat 100 0 0.522 0.294 −0.026 0.495 1.460
theta_hat_t_stats 100 0 2.117 0.874 −0.138 2.145 4.145
R2_demand 100 0 0.293 0.138 −0.167 0.305 0.528
R2_supply 100 0 0.976 0.004 0.962 0.977 0.985
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.032 0.204 9.599 10.041 10.669
alpha1_hat 100 0 1.042 0.942 −1.293 1.059 3.488
alpha2_hat 100 0 1.005 0.111 0.657 1.021 1.303
alpha3_hat 100 0 1.004 0.041 0.910 1.006 1.118
gamma0_hat 100 0 0.977 0.337 0.078 1.062 1.889
gamma1_hat 100 0 1.048 1.094 −1.895 1.142 4.479
gamma2_hat 100 0 1.000 0.052 0.896 0.994 1.127
gamma3_hat 100 0 1.002 0.040 0.874 1.001 1.100
theta_hat 100 0 0.499 0.118 0.178 0.490 0.891
theta_hat_t_stats 100 0 4.918 0.868 2.051 4.879 6.597
R2_demand 100 0 0.313 0.053 0.145 0.312 0.413
R2_supply 100 0 0.975 0.002 0.968 0.975 0.980
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.032 0.154 9.640 10.025 10.500
alpha1_hat 100 0 1.070 0.659 −0.595 0.976 2.432
alpha2_hat 100 0 1.003 0.079 0.770 1.007 1.191
alpha3_hat 100 0 1.005 0.031 0.933 1.000 1.081
gamma0_hat 100 0 0.985 0.240 0.390 1.013 1.612
gamma1_hat 100 0 1.014 0.855 −0.839 0.989 3.075
gamma2_hat 100 0 1.005 0.034 0.920 1.004 1.077
gamma3_hat 100 0 1.001 0.031 0.911 1.001 1.085
theta_hat 100 0 0.495 0.082 0.239 0.488 0.665
theta_hat_t_stats 100 0 7.011 0.952 4.147 6.976 9.101
R2_demand 100 0 0.313 0.042 0.214 0.318 0.400
R2_supply 100 0 0.975 0.002 0.971 0.975 0.980
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.987 0.095 9.744 9.980 10.205
alpha1_hat 100 0 0.971 0.444 −0.180 0.945 2.023
alpha2_hat 100 0 0.998 0.049 0.888 0.999 1.115
alpha3_hat 100 0 0.996 0.021 0.948 0.993 1.054
gamma0_hat 100 0 1.028 0.139 0.716 1.005 1.398
gamma1_hat 100 0 1.070 0.436 0.167 1.053 2.048
gamma2_hat 100 0 0.996 0.021 0.944 0.998 1.064
gamma3_hat 100 0 0.996 0.021 0.948 0.994 1.054
theta_hat 100 0 0.492 0.045 0.365 0.494 0.600
theta_hat_t_stats 100 0 11.059 0.780 8.879 11.120 12.860
R2_demand 100 0 0.320 0.025 0.255 0.319 0.366
R2_supply 100 0 0.975 0.001 0.973 0.975 0.977

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.905 0.857 7.903 9.839 13.046
alpha1_hat 100 0 1.229 19.905 −56.477 −1.308 46.467
alpha2_hat 100 0 4.838 2.291 −0.843 5.057 11.967
alpha3_hat 100 0 0.994 0.178 0.653 0.978 1.503
gamma0_hat 100 0 1.809 1.280 −2.402 1.841 3.859
gamma1_hat 100 0 −6.955 141.547 −782.027 1.434 985.368
gamma2_hat 100 0 0.915 0.209 0.565 0.887 1.456
gamma3_hat 100 0 0.922 0.188 0.590 0.915 1.770
theta_hat 100 0 0.521 3.992 −29.106 0.393 19.570
theta_hat_t_stats 100 0 0.907 0.994 −2.764 1.137 2.712
R2_demand 100 0 0.355 0.287 −1.042 0.408 0.736
R2_supply 100 0 0.980 0.006 0.960 0.981 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.060 0.675 8.425 10.004 12.044
alpha1_hat 100 0 0.282 13.587 −33.200 −1.095 43.051
alpha2_hat 100 0 5.147 1.474 0.676 5.129 9.030
alpha3_hat 100 0 0.999 0.156 0.745 0.993 1.553
gamma0_hat 100 0 1.454 0.954 −1.688 1.478 3.492
gamma1_hat 100 0 0.585 22.873 −183.389 2.925 33.096
gamma2_hat 100 0 0.945 0.147 0.609 0.922 1.355
gamma3_hat 100 0 0.945 0.138 0.627 0.958 1.285
theta_hat 100 0 0.448 0.464 −0.190 0.390 4.004
theta_hat_t_stats 100 0 1.325 0.762 −0.834 1.379 2.948
R2_demand 100 0 0.312 0.260 −0.988 0.370 0.586
R2_supply 100 0 0.977 0.006 0.943 0.977 0.985
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.960 0.473 8.789 9.966 11.243
alpha1_hat 100 0 −0.095 9.647 −19.686 0.279 20.684
alpha2_hat 100 0 5.058 1.106 2.841 4.852 8.631
alpha3_hat 100 0 0.994 0.114 0.770 0.985 1.251
gamma0_hat 100 0 1.135 0.753 −1.716 1.138 2.706
gamma1_hat 100 0 −0.003 11.338 −35.419 1.141 24.329
gamma2_hat 100 0 0.982 0.112 0.741 0.981 1.341
gamma3_hat 100 0 0.985 0.119 0.754 0.959 1.331
theta_hat 100 0 0.511 0.251 0.026 0.466 1.389
theta_hat_t_stats 100 0 2.412 0.896 0.133 2.319 4.316
R2_demand 100 0 0.342 0.130 −0.090 0.361 0.561
R2_supply 100 0 0.975 0.005 0.956 0.976 0.984
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.016 0.184 9.555 10.003 10.435
alpha1_hat 100 0 1.571 3.895 −7.121 1.534 10.478
alpha2_hat 100 0 4.965 0.453 3.772 4.948 5.792
alpha3_hat 100 0 1.003 0.043 0.913 1.002 1.110
gamma0_hat 100 0 1.071 0.273 0.276 1.058 1.724
gamma1_hat 100 0 0.459 4.648 −9.589 0.220 12.321
gamma2_hat 100 0 0.993 0.040 0.899 1.000 1.122
gamma3_hat 100 0 0.987 0.042 0.877 0.986 1.110
theta_hat 100 0 0.497 0.095 0.216 0.497 0.692
theta_hat_t_stats 100 0 5.443 0.885 2.975 5.392 7.455
R2_demand 100 0 0.341 0.053 0.182 0.347 0.431
R2_supply 100 0 0.975 0.002 0.969 0.975 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.005 0.139 9.680 10.007 10.451
alpha1_hat 100 0 1.166 2.547 −3.731 0.902 6.745
alpha2_hat 100 0 4.994 0.341 4.223 5.031 5.811
alpha3_hat 100 0 1.004 0.034 0.927 1.005 1.089
gamma0_hat 100 0 1.030 0.201 0.525 1.040 1.616
gamma1_hat 100 0 0.478 3.330 −10.025 0.656 9.538
gamma2_hat 100 0 0.996 0.028 0.903 0.995 1.070
gamma3_hat 100 0 0.995 0.029 0.882 0.996 1.044
theta_hat 100 0 0.505 0.081 0.305 0.507 0.718
theta_hat_t_stats 100 0 7.807 0.977 4.737 7.810 10.650
R2_demand 100 0 0.344 0.042 0.237 0.346 0.452
R2_supply 100 0 0.974 0.001 0.970 0.974 0.976
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.085 9.800 10.005 10.220
alpha1_hat 100 0 0.678 1.526 −2.974 0.854 3.926
alpha2_hat 100 0 5.032 0.195 4.544 5.025 5.640
alpha3_hat 100 0 0.998 0.018 0.954 1.001 1.045
gamma0_hat 100 0 0.997 0.142 0.656 0.997 1.308
gamma1_hat 100 0 1.449 1.883 −3.388 1.445 5.951
gamma2_hat 100 0 1.000 0.021 0.951 1.000 1.051
gamma3_hat 100 0 1.000 0.019 0.962 0.997 1.048
theta_hat 100 0 0.492 0.044 0.387 0.490 0.622
theta_hat_t_stats 100 0 12.194 0.870 10.371 12.196 14.320
R2_demand 100 0 0.348 0.023 0.293 0.348 0.397
R2_supply 100 0 0.974 0.001 0.971 0.974 0.976

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.952 1.012 8.115 9.864 13.583
alpha1_hat 100 0 −6.081 81.140 −175.858 −5.806 224.524
alpha2_hat 100 0 20.447 8.570 −2.178 20.765 39.275
alpha3_hat 100 0 0.966 0.235 0.550 0.939 1.762
gamma0_hat 100 0 2.037 1.210 −2.040 2.072 5.378
gamma1_hat 100 0 −165.805 1276.000 −12591.352 −2.780 171.147
gamma2_hat 100 0 0.890 0.192 0.481 0.875 1.523
gamma3_hat 100 0 0.902 0.194 0.432 0.879 1.413
theta_hat 100 0 1.322 7.290 −0.592 0.354 71.823
theta_hat_t_stats 100 0 0.873 1.076 −2.113 0.850 3.381
R2_demand 100 0 0.319 0.389 −1.852 0.434 0.806
R2_supply 100 0 0.980 0.007 0.948 0.982 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.919 0.624 8.551 9.892 12.815
alpha1_hat 100 0 3.180 50.544 −105.249 3.688 130.740
alpha2_hat 100 0 19.360 6.359 5.189 18.479 43.643
alpha3_hat 100 0 1.000 0.152 0.649 0.982 1.449
gamma0_hat 100 0 1.568 0.671 −0.425 1.573 3.326
gamma1_hat 100 0 −8.909 62.845 −251.951 3.967 104.542
gamma2_hat 100 0 0.938 0.110 0.660 0.937 1.261
gamma3_hat 100 0 0.937 0.128 0.649 0.943 1.228
theta_hat 100 0 0.481 0.376 −0.128 0.429 1.981
theta_hat_t_stats 100 0 1.491 0.884 −0.391 1.445 3.865
R2_demand 100 0 0.351 0.274 −1.872 0.380 0.676
R2_supply 100 0 0.977 0.005 0.964 0.977 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.013 0.473 8.633 9.987 11.164
alpha1_hat 100 0 0.905 38.348 −127.127 7.208 71.531
alpha2_hat 100 0 20.126 4.806 9.769 19.467 38.683
alpha3_hat 100 0 1.013 0.100 0.800 1.018 1.314
gamma0_hat 100 0 1.194 0.717 −2.181 1.196 2.553
gamma1_hat 100 0 −6.568 38.363 −102.316 −4.638 71.359
gamma2_hat 100 0 0.970 0.108 0.720 0.972 1.277
gamma3_hat 100 0 0.982 0.110 0.742 0.986 1.428
theta_hat 100 0 0.522 0.266 0.036 0.496 1.611
theta_hat_t_stats 100 0 2.438 0.882 0.208 2.473 4.863
R2_demand 100 0 0.326 0.140 −0.064 0.359 0.547
R2_supply 100 0 0.975 0.004 0.952 0.976 0.984
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.192 9.555 9.974 10.513
alpha1_hat 100 0 2.092 14.772 −39.193 3.266 45.123
alpha2_hat 100 0 19.835 1.710 14.689 19.769 23.925
alpha3_hat 100 0 1.002 0.042 0.916 0.996 1.100
gamma0_hat 100 0 1.063 0.293 0.125 1.056 2.002
gamma1_hat 100 0 2.248 16.270 −40.032 2.669 43.902
gamma2_hat 100 0 0.992 0.044 0.878 0.993 1.127
gamma3_hat 100 0 0.999 0.048 0.894 0.994 1.142
theta_hat 100 0 0.486 0.094 0.261 0.491 0.712
theta_hat_t_stats 100 0 5.404 0.857 3.233 5.410 7.154
R2_demand 100 0 0.356 0.048 0.230 0.362 0.464
R2_supply 100 0 0.974 0.002 0.968 0.974 0.979
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.192 9.555 9.974 10.513
alpha1_hat 100 0 2.092 14.772 −39.193 3.266 45.123
alpha2_hat 100 0 19.835 1.710 14.689 19.769 23.925
alpha3_hat 100 0 1.002 0.042 0.916 0.996 1.100
gamma0_hat 100 0 1.063 0.293 0.125 1.056 2.002
gamma1_hat 100 0 2.248 16.270 −40.032 2.669 43.902
gamma2_hat 100 0 0.992 0.044 0.878 0.993 1.127
gamma3_hat 100 0 0.999 0.048 0.894 0.994 1.142
theta_hat 100 0 0.486 0.094 0.261 0.491 0.712
theta_hat_t_stats 100 0 5.404 0.857 3.233 5.410 7.154
R2_demand 100 0 0.356 0.048 0.230 0.362 0.464
R2_supply 100 0 0.974 0.002 0.968 0.974 0.979
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.089 9.789 10.003 10.203
alpha1_hat 100 0 0.465 6.336 −19.340 0.529 15.995
alpha2_hat 100 0 20.081 0.789 18.157 20.126 22.977
alpha3_hat 100 0 1.002 0.019 0.961 1.000 1.045
gamma0_hat 100 0 1.016 0.135 0.720 1.019 1.340
gamma1_hat 100 0 0.197 7.536 −23.849 0.396 15.744
gamma2_hat 100 0 0.999 0.020 0.950 0.998 1.050
gamma3_hat 100 0 0.998 0.022 0.948 0.997 1.052
theta_hat 100 0 0.500 0.046 0.418 0.498 0.625
theta_hat_t_stats 100 0 12.623 0.892 10.564 12.657 14.663
R2_demand 100 0 0.356 0.021 0.296 0.357 0.399
R2_supply 100 0 0.974 0.001 0.970 0.974 0.976

Plot

iv_polynomial

50

## Warning: Removed 4 rows containing missing values (`geom_col()`).

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